## Abstract

Animal legs are capable of a tremendous breadth of distinct dynamic behaviors. As robots pursue this same degree of flexibility in their behavioral repertoire, the design of the power transition mechanism from joint to operational space (the leg) becomes increasingly significant given the limitations current actuator technology. To address the challenges of designing legs capable of meeting the competing requirements of various dynamic behaviors, this paper proposes a technique which prioritizes explicitly encoding a set of dynamics into a robot’s leg design, called dyno-kinematic leg design (DKLD). This paper also augments the design technique with a method of evaluating the suitability of an individual leg’s workspace to perform dynamic behaviors, called the effective dynamic workspace (EDW). These concepts are shown to effectively determine optimal leg designs within a set of three, increasingly complex, case studies on different robots. These new legs designs enable a 5 kg robot to climb vertical surfaces at 3 Hz, allow a 60 kg robot to efficiently perform a range of behaviors useful for navigation (including a run at 2 m/s), and endow a small quadrupedal robot with all of the necessary behaviors to produce running and climbing multimodality. This design methodology proves robust enough to determine advantageous legs for a diverse range of dynamic requirements, leg morphologies, and cost functions, therefore demonstrating its possible application to many legged robotic platforms.

## 1 Introduction

Legged animals have a wide variety of leg morphologies which perform a diverse array of locomotive behaviors. These designs range from the multisegmented, biramous, and uniramous limbs produced by arthropods exoskeletons’ [1] to the variations of the digitigrade, plantigrade, and unguligrade musculo-skeletal leg structures used by many terrestrial vertebrates [2]. These diverse leg morphologies have adapted and evolved over millennia in order to better suit the locomotive tasks of the animal utilizing them. The leg design of kangaroos, for example, was studied by Alexander and Vernon [3] in which it was found that the musculo-skeletal structure was well suited for the task of continuous hopping. Animals can also have legs well developed for the challenges of multimodality such as the legs of leopard frogs being well developed for both hopping and swimming [4].

Many leg designs in robotics borrow from biology and select bio-mimetic leg designs. Robots such as Boadicea [5] and Robot V [6] in the horizontal domain or Stickybot [7] in the vertical have emulated the workspace and kinematic structure found on animals. However, as electro-mechanical actuators can not yet match the sophistication of the muscular systems in animals, power limitations and precise force application prevent these systems from directly mapping the animal’s control. This approach shifts the emphasis from the leg design and instead places the most rigorous design challenge on the robots’ control. Control-centralized design takes a flexible, often a simplified bio-mimetic, design and focuses designing controllers on getting the most out of the selected leg. Examples of this approach include high degree of freedom humanoids [8,9], hexapods [10–12], and quadrupeds [13,14]. Although robots which use this technique have been shown to generate a variety of behaviors [15] they are typically unable to generate the necessary foot forces and velocities to locomote at high speed or in multiple domains.

More mechanism-centralized approaches also exist in which the designer focuses on the development of the power transmission and/or the paths the leg may take while moving. These robots attempt to generate large thrust vectors at the ground and develop mechanisms to best leverage the actuators and maximize the power transmission in the direction of motion. Examples of this include the tail and thrusters on the hopping robot Salto-1p [16], the powertrain on MABEL [17], and the optimization of the hydrolic actuators on HyQ [18]. In a similar vein some designs look to the inverse of the problem and attempt to improve transparency in the leg mechanism to allow for force sensing and control using the actuators. Such as the low impedance leg and brushless motors designed on robots like Minitaur [19] and Cheetah [20]. Other mechanism-centric robotic leg designs take into account a desirable foot path for locomotion and intentionally reduce the dimensionality of the leg in order to achieve improved mechanical advantage such as the variable leg stiffness on the c-legs in Rhex [21], the crank path for climbing on RISE [22], and foot path based kinematic design on DynaRoACH [23]. Although both the control and mechanism-centered design methods have produced robots with impressive locomotive performance in a range of modalities, individual robots have generally been restricted to a limited subset of locomotive behaviors, in part because legs have not been designed to handle the competing demands of different dynamics.

Design techniques for robotic legs in general consider the leg workspace as having a homogeneous set of constraints, however individual locomotive behaviors each have their own unique requirements and may only use a subsection of the total workspace (i.e., upright running holds the foot under the hip, or opening a door may push the foot upward and infront of the body). This suggests that it may be possible to improve or develop a leg design for multiple behaviors by targeting workspace regions and optimizing based on the dynamic constraints required within that region. In this paper, we propose a design technique which does this by explicitly encoding a set of desired locomotive behaviors (target dynamics) into the design process of a robot’s leg morphology (kinematics) by observing the constraints and performance of these behaviors as critical design elements. This method is representative of a dynamics-centralized design approach, which may implicitly obtain both the mechanism flexibility and precise mechanical advantage improvements of the other approaches by focusing on design to capture the dynamics.

This paper begins with a description of this “Dyno-Kinematic” (DK) approach to leg design and workspace evaluation. Following this three distinct case studies are presented. The first of these studies uses this concept to add dynamic climbing capabilities to a robot adapted from the Minitaur platform called BOBCAT.^{2} The second is used to enable diverse walking and running behaviors on a large legged platform LLAMA. The final study is used to develop a leg morphology for a future, small-scale, multimodal, quadrupedal robot.

## 2 Dyno-Kinematic Leg Design Method

Our design goal is to create legs capable of producing the desired locomotion (or target) high-energy behaviors of a robot as expressed in operational space. In this paper, we describe the dyno-kinematic leg design (DKLD) procedure and introduce a method of visualizing the workspace called the effective dynamic workspace (EDW) which explicitly consider the requirements of desired locomotive behaviors.

Dyno-kinematic leg design is a procedure that may be used in design which encodes the requisite locomotion dynamics (positions, velocities, and forces) into the design process for limb morphology. As shown in Fig. 1, the principle step in this process is constructing a set of target dynamics for a collection of desired locomotive behaviors. Following this a leg type is selected which has a set of kinematic equations and a Jacobian parameterized by physical linkage sizes. After which an assortment of physical constraints associated with the locomotion is defined in both joint and operational spaces. The condition of these yields the possible subset of leg designs which can instantiate the full set of target dynamics. This set can then be optimized to select candidate legs for design.

The effective dynamic workspace is the subspace of the kinematic workspace or foot positions contained within the workspace of a leg design that can achieve the desired locomotion dynamics of the leg. As shown in Fig. 1, the EDW can act as a visualization tool to evaluate the performance of a particular leg design’s workspace under the target dynamics (or an approximation of the target dynamics) and the locomotive constraints. Each parameter set within a leg type has its own EDW which can range in size from the full usable workspace to a null set.

The subcomponents which form the design procedure of the DKLD and define the EDW are outlined below.

### 2.1 Target Dynamics.

Many diverse behaviors can be exhibited in legged locomotion (such as walking, crawling, jumping, and dynamic climbing), each of which are characterized by different distinguishing traits and use distinct parts of a leg’s workspace. In order to define these behaviors for leg design evaluation, a set of target dynamics need to be constructed. These target dynamics are made up of two parts: a motion profile (the set of positions and velocities of the foot relative to the hip) and a force profile (the set of forces generated by the foot) which are produced during locomotion. In order to facilitate freedom in the construction of behaviors the target dynamics are produced agnostic of leg morphology.

The target dynamics are attainable through various methods. Reduced order models, such as the SLIP [25,26] and Full-Goldman (FG) [27] models, may be used to generate the forces and motions of the leg in foot space. These models have been shown to effectively capture the ground reaction force magnitudes and velocity profiles of both running and climbing animals. They allow numerical integration of analytically derived equations of motion that can be used represent of a wide range of behaviors [28,29].

Other methods for generating the target dynamics include physics-based simulations (such as Simscape™, and ADAMs™) which may be used to capture more complex dynamic behaviors beyond the scope of simplified models. Similarly, the target dynamics may be recorded experimentally from the testing on functional robots or animals which the designer is attempting to emulate. These methods allow direct recording the differing forces of highly complex multilegged systems and may be used to capture the nuanced variation in locomotion dynamics which occur during disturbance recovery. Using this method on a functioning robot has the potential added benefit of improving leg design for operational robots without invalidating previously developed behaviors. However, as a note, target dynamics produced on a robot are not guaranteed to be preserved using the DKLD if the locomotion is altered or inhibited by physical limitations, such as actuator saturation. Many, if not all, of these methods require some kind of control to generate the targets, as such the desired control strategy maybe integrated with your mechanism design. The composition of the target dynamics can, and often should, contain multiple behaviors which span a sufficient range of tasks to capture the critical elements of locomotion.

### 2.2 Kinematics.

After the desired behaviors have been constructed, a set of leg kinematics can be imposed upon them as a mapping from the operational (foot) space into the joint (motor) space. The inverse leg kinematics can be used to evaluate the motor inputs which may (or may not) achieve the motion profile and may also be used to transmit the force profile through the linkages to evaluate leg stresses. The Jacobian is used to map the force profile to motor torques.

The kinematic equations vary on both leg type (i.e., the set of connected joints) and the linkage parameters (the set of physical rigid lengths and angles which characterize the leg). Both of these can have a profound effect on the leg’s capabilities. The leg type changes the fundamental mechanism for power transmission, alters the degrees of freedom of the leg, and can alter the number of linkage parameters needed to define the kinematics equations. The linkage parameters (even with small variations) can have a significant impact on both workspace size and a leg’s performance.

### 2.3 Constraints.

Constraints are added in order to bound the behaviors and kinematics to results which are physically achievable by the robot. The most ubiquitous of these constraints are given by the actuator model and the kinematic limits. These constraints serve only as pass/fail criteria of the target dynamics. The leg kinematics specifically may have multiple factors that may result in failure of the motion profile, such as the motion profiles being outside the workspace bounds of a tested leg design, or prohibiting linkage contact with the ground other than with the foot.

Other constraints may bound, augment, or place special requirements on the target dynamics or kinematics. Foot placement may experience special requirement such as securing an appropriate incident angle (particularly needed in climbing for attachment mechanisms) or ensuring sufficient frictional contact with the ground. As the lower-dimensional models used for the target dynamics may not capture all of the requisite forces in locomotion, some augmentation to these models may be necessary, in the form of orthogonal models or manipulation of the motion profile in the workspace, to accurately enforce viable motions in three dimensions.

### 2.4 Design Selection Cost Functions.

By applying the target dynamics and constraints to a selected leg type across a range of linkage parameters, a subset of legs which are capable of the desired behaviors can be determined within the *n*-dimensional space of candidate design parameters. In order to select from these potential candidate designs an optimization can be used to select a preferred design via a desired performance metric. Possible metrics include peak torque, efficiency, actuator work ratio, workspace size, etc. Notably, the “power quality” metric discussed in Abate et al. [30] (which is used to minimize actuator antagonism for a single behavior) can be computed via the first two steps in the DKLD process and may be utilized as an objective function here. As the space has already been shown to satisfy the design constraints, any of these metrics allow the designer to select leg parameters with desirable design features that are ensured to be capable of performing the locomotive tasks.

### 2.5 Effective Dynamic Workspace.

Dyno-kinematic leg design assists in selecting legs which can best instantiate the desired set of locomotive behaviors. Utilizing the same component parts of the design procedure, a conservative estimate of a legs performance may be visually represented in operational space, forming the effective dynamic workspace Fig. 1. Given the two- or three-dimensional foot workspace a leg design may have within its reachable space there exists a smaller effective space which may reasonably be utilized for a locomotive behavior. The EDW is populated by points in the total workspace which can generate the target dynamics (or a conservative estimate of them) of that local region of the space subject to the design constraints. This visualization can subsequently be enriched by color coding the viable regions with a performance metric. By observing this space, a more accurate representation of functional workspace size and localized performance improvements can be depicted through variations to the kinematic parameters.

## 3 Design Case Studies

The following sections of this paper consist of three distinct case studies in which the concepts of DKLD and EDW are used to select and evaluate robotic leg designs. The studies take on the task of generating high performance legs for three robots which must be tuned for diverse locomotive behaviors and varying design goals. These studies pose design challenges for optimizing for differing leg morphologies and for target dynamics consisting of both a single and multiple behaviors. Initially, the design process is implemented on only one leg morphology and one behavior composing the target dynamics. The second study augments this from a single behavior to multiple behaviors on a different morphology. The final and most complex study simultaneously tests several candidate leg morphology’s and a range of target dynamics, including walking, running, pouncing, and high-speed climbing.

### 3.1 General Procedure and Optimization Algorithm.

In the following case studies, the target dynamics are formed using reduced order models utilizing differing control schemes to generate diverse behaviors (walk, run, climb, etc.). A set or sets of leg morphologies are selected for each case study and parameterized to a space of fixed linkage lengths and angles which define the kinematics. Conditions are then articulated which define the behavioral constraints, physical constraints, and the boundaries of the linkage parameters. Design selection is performed using optimization as the range of possible leg morphologies which may achieve a particular behavior can exist within a high-dimensional space.

When optimizing a linkage for multiple behaviors the final cost used by the algorithm is the Euclidean norm of all the costs of the individual behavioral modes shown in Eq. (1). If a leg cannot perform the behavior within the boundaries of the constraints that configuration is marked as failing and is given a high cost. The algorithm converged quickly, generally taking fewer than 3 min in even the most complex optimization problem studied here.

## 4 Case Study 1: BOBCAT Climbing

BOBCAT’s precursor, Minitaur [19] is a 5.45 kg quadrupedal robot which can generate walking and running behaviors. This robot utilizes a symmetric, coaxial 5-bar leg morphology with two degrees of freedom to produce a toroidal workspace while placing the actuators near the center of mass. The motors are direct drive which produces low mechanical impedance, facilitating the generation of a virtual spring corresponding to the proportional gain of the leg position controller. Using an optimized feed-forward trajectory controller this robot has been shown to be capable of running at 7.1 body lengths per second [33].

Although this robot is capable of impressive performance during horizontal ground locomotion and can almost climb vertically, initial experimentation showed that Minitaur was incapable of climbing dynamically. The motors could only generate sufficient torque to produce very slow locomotion upward using a crawling gait. This motivated the creation of a newly designed multimodal platform BOBCAT and provided an excellent test bed for design via DKLD.

### 4.1 BOBCAT Platform.

Figure 2 shows BOBCAT with the 5-bar linkage morphology with coaxially aligned motors used on Minitaur [19]. In adapting BOBCAT from Minitaur, to better facilitate climbing, the following design objectives were considered: weight reduction, lowering the center of gravity, and expanding the available workspace when climbing. The new body frame, made of ABS plastic with aluminum hip brackets, resulted in a reduction of overall mass to 5.0 kg. The new body and hip structures relocate the body 51 mm lower relative to the hips (lowering the CG by 17 mm during standing). The unimpeded leg workspace was extended 23 mm toward the body through redesign of the hip-to-body fastening interface. To maintain the direct drive capabilities, the motors, leg linkages, and related electronics were directly migrated from Minitaur to BOBCAT.

A simple, ABS plastic, multimodal end effector (Fig. 2(c)) was designed for the leg which houses an array of three independently sprung claws which were developed based off of the microspine arrays used on robots such as BOB [28] and RiSE [22]. Figure 2(d) shows the claws, called macrospines, which enable directional adhesion to the substrate with the rapid attachment and detachment needed to produce dynamic climbing. Unlike the previously used microspines, stronger elastomeric material was used (40A durometer urethane rubber) and the thickness of each toe was increased to 4 mm. The tensile strength of these macrospines was increases from its predecessor by over 16-fold, therefore a single toe is strong enough to support the robot’s full weight.

## 5 BOBCAT Climbing: Modified Full-Goldman

While the desired dynamic behaviors for BOBCAT include both high-speed running and climbing, both Minitaur and initial testing with BOBCAT demonstrated walking and running required relatively low motor effort using both the current leg design as well as variants [34]. Thus, for this first study the target dynamics being considered for the DKLD are constructed solely from a model of dynamic climbing. The FG model has been shown to effectively capture the ground reaction force magnitudes and center of mass profiles of climbing animals [27] and dynamic climbing bipedal [35,36] and quadrupedal [37] robots. The model consists of a central mass supported by a linearly actuated arm in line with a linear spring pinned to the world frame at the foot. The model climbs by transitioning between two arms held at a fixed angular offset apart.

*D*. The model consists of a point mass

*M*centered along the line between the front legs. Because the robot’s control is assumed to enforce a trot,

*M*makes only up half the mass of the robot (as the load is considered to be evenly distributed between the front and rear legs). The point mass is being acted on by gravity with acceleration

*g*. The spring stiffness in line with the leg, produced by the position control of the low impedance motors, is denoted by the parameter

*k*. The angle of the arm

*θ*with the vertical and the spring deflection

*s*are determined through the dynamics. The actuated parameter in this model is the leg length

*L*which originates at a distance

*D*, or half the robots body width, from the point mass

The equations of motion, Eqs. (2) and (3), were derived using Lagrange techniques and can be used to compute the ground reaction forces in the plane of the wall during steady-state climbing. The portion of the force profile within this plane is computed for a single step. In order to generate climbing behaviors a simple feed-forward trajectory was used which sets the leg length *L* to retract linearly at a fixed velocity.

The modified FG model only considers forces and motions in the horizontal plane (which corresponds to only one dimension of the BOBCAT’s leg workspace) therefore the model must be augmented to fully construct the force and motion profiles. In order to complete this a supplementary quasi-static model is developed which is acting in the sagittal plane normal to the wall, shown in Fig. 3(b). The distance *D*_{body} refers to the distance from the wall at which the leg is retracting. *M*_{g} refers to the instantaneous force of gravity experienced by the robots center of mass based on the angle of the body *θ* from the modified FG. Forces *F*_{1} and *F*_{2} represent forces that the robot applies on the wall counteract external forces such as those induced by gravity. For the purposes of this study, only the length *L*_{leg1}, *L*_{leg2}, (which are set by the target dynamics) and *D*_{body} are varied based on the kinematic parameters of the leg. A description of the how *D*_{body} set is defined in the constraints section.

## 6 Generic 5-bar

*M*

_{1}and

*M*

_{2}, are measured globally from a line directly in front of the leg parallel to the surface. The primary links

*L*

_{1}and

*L*

_{3}correspond to motor angles

*M*

_{2}and

*M*

_{1}, respectively. The secondary linkages

*L*

_{2}and

*L*

_{4}connect to the primary linkages and are pinned to each other at the toe. These kinematics do not explicitly consider the toe extension which extends linearly from

*L*

_{4}on the physical robot. The angles

*α*and

*β*relate to the angle of virtual leg

*ϕ*such that

*R*and leg angle

*ϕ*.

*L*

_{B}(the length between the knees) and

*ψ*

_{34}(the angle between

*L*

_{3}and

*L*

_{4}) are defined by

*α*and the length of the virtual leg

*R*were determined through the law of cosines, shown below.

*ϕ*is determined by inverting Eq. (4). The inverse kinematics can be found using the law of cosines from Eq. (10) (identically used to find

*β*) which when combined with the known angle

*ϕ*produce the motor angles

*M*

_{1}and

*M*

_{2}(Eqs. (4) and (5))

The Jacobian (derived from the inverse kinematics) used to map the velocity and force profile in the foot space to joint velocities and torques is shown in Eq. (11).

## 7 BOBCAT Constraints

The robot’s locomotion is primarily constrained by the motor model, in this case, a linear model of the brushless U8 motor from T-motor (*τ*_{stall} = 7.5 N m and *ω*_{No Load} = 84 rad/s) [38]. Additional constraints used for dynamic climbing including ensuring attachment, minimizing the impact on out-of-plane dynamics, and balancing the pitch-back moment. To ensure attachment, the foot space must be below the hip (further than the frame) and be no shorter than inner primary link length, with the angle of the attachment mechanism constrained to be within a fixed angular deflection of vertical. In this case, the claws used to attach to the substrate require a certain range of incident foot angles between $+15\u2218$ and $\u221210\u2218$ from parallel with the substrate in order to allow directional adhesion (determined via hand testing). To minimize the impact on the out-of-plane dynamics, the trajectory is set to be continuously retracting and maintains a fixed distance away from the wall throughout the stroke. This is held constant to regulate the pitch-back moment created when the body mass is far away from the wall, and minimize the total amount of energy expended by regulating body distance from the wall. The forces produced from the pitch-back moment are computed from a quasi-static model (shown in Fig. 3(b)) and used to augment the force of the modified FG model.

The range of possible linkages (shown in Table 1) was constrained to a feasible design space with limits determined by the robot’s dimensions and the goal of minimally changing the robots mass. The space is centered around the initial symmetric design point with the primary links set to 10 cm and the secondary links set to 20 cm. The lower bound of the primary links, *L*_{1} and *L*_{3}, was set to 6 cm to maintain sufficient clearance with the motor radius while the upper bound is set only 1 cm larger than the nominal configuration as lengthening the moment arm would result in interference with the frame and surface.

## 8 BOBCAT: Simulation

Leg design for BOBCAT was performed in a matlab simulation environment. To construct the target dynamics, the equations of motion for the modified FG model were numerically integrated using matlab’s ODE45 command. The physical parameter values for this simulation can be found in Table 1. The orthogonal quasi-static model causes variation of the position profile based on length *L*_{3} (to minimize the pitching moment). This couples with the attachment angle constraint bounding the incident angle to a small range, to cause the usable workspace region for climbing to be highly variable based on the kinematic parameters. In order to increase the number of achievable linkage configurations within the constraints a parameter sweep of the leg actuation in the modified FG model was performed to compose the target dynamics. The FG model parameters used in the sweep are shown in Table. 1. Nominal leg lengths were swept from 15 cm to 25 cm at an increment of 0.1 cm while the commanded retraction maintained the minimum desired stroke length of 11 cm.

*L*

_{3}+ 1 cm away from the hip (the value of

*D*

_{body}in Fig. 3) and using the closest nominal leg length to the center of this retraction. With this relationship between the target dynamics and kinematics parameters the selection of an optimal leg for climbing is then made using the optimization algorithm to complete the dyno-kinematic leg design

A cost function was chosen that minimized peak torque (*τ*_{pk}), Eq. (12), during locomotion in order to select designs which could better avoid motor limits. This simulated optimal design for this study is additionally validated experimentally on the BOBCAT platform.

### 8.1 Optimum.

The particle swarm linkage optimization selected a leg with lengths *L*_{1} = 6.25 cm, *L*_{2} = 25 cm, *L*_{3} = 6.5 cm, and *L*4 = 24.75. The optimization converged to a region with the primary links near the minimum allowable length and the secondary links extending to or near the maximum. The optimum converged toward a near symmetric linkage which reduced the peak torque by more than half compared to the nominal leg. Additionally, the work ratio (the ratio between the average work done by the inner motor relative to the total work) reduced from 73% the in the nominal configurations to 53%, better distributing the load between the two motors.

The EDW, Fig. 5, shows the workspace of the nominal leg (left) compared with that of the optimum leg (right). The gradient corresponds to the maximum motor torque with a single leg supporting the full weight of the robot (worst case scenario) computed from the Jacobian. The maximum motor torque is an effective representation of the robustness of that region of the workspace to instantaneous variations in GRF. The nominal trajectory of the foot relative to the body is shown in black. While the total workspace area of the optimum leg is 32% smaller, the effective dynamic workspace is increased by 210%. The motors in the nominal configuration stall shortly into the trajectory while the optimum leg reduces the average requisite torque throughout the trajectory by more than half allowing it to achieve the full stroke within its kinematic limitations.

## 9 Experimental Validation

Two distinct leg configurations (the nominal leg used on minitaur and a near-optimal symmetric leg) were experimentally tested on the BOBCAT platform in order to validate the DK design methodology. On board computing constraints limited the speed at which the kinematics of the asymmetric 5 bar could be computed, so only symmetric linkages were selected for experimental testing, with the resulting design point having *L*_{1} = *L*_{3} = 6.5 cm and *L*_{2} = *L*_{4} = 25 cm. While this configuration’s performance was not quite optimal, the change in peak torque is less than 2% from the optimal. Experimentation consisted of tests varying the driving frequency of the leg from 1 to 3 Hz while holding the linear and torsional stiffnesses constant between leg morphologies. Although the design procedure here focus only on design for vertical locomotion, tests were performed for both climbing and running in order to ensure the leg could still produce the necessary behaviors in the horizontal domain. Speed and motor effort data were recorded during experimentation.

Climbing experiments were performed using the macro-spines described in the section “BOBCAT Platform” attaching to a quarter inch wire mesh grid backed by carpet. Preliminary testing on a purely carpet wall resulted in inconsistent attachment, detachment, and deformation of the substrate. The wire mesh grid was used to support the weight of the platform and thus reduce the chance for surface failures while still allowing for rapid attachment and detachment. In addition to the reinforced surface, the robot was constrained in the out-of-plane direction using a nylon strap to regulate the pitch and roll dynamics which require precise tuning (which is outside the scope of this paper).

Running experiments were recorded using Vicon motion tracking system while climbing performance was measured using high-speed camera footage. To ensure maximum power electrical power input, all experiments were run on battery which was kept above 95% of the maximum voltage.

### 9.1 Control.

The controller implementation on BOBCAT leverages several of the benefits of direct drive actuation, namely tunable torsional spring stiffnesses and force approximation via motor effort, which have previously been implemented on Minitaur [39,40]. Since BOBCAT is designed to generate multimodal behaviors, the controller design was required to be able to operate both directly below the hips as well as almost directly in front of or behind the hips.

Previous running work on Minitaur implemented a smoothed feed-forward trajectory based around a reduced set of design points [33,41]. As such, a trajectory-based approach with smooth foot paths generated via Bezier curves defined by the points in Fig. 6 implemented a version of impulse control where the prescribed stance trajectory extends into the surface [42] was used for both running and climbing. Since running is primarily controlled in RP space, the torsional springs at the hips are transformed into a virtual prismatic and torsional spring pair via the manipulator Jacobian.

While the controllers for running and climbing are both implemented via a feed-forward trajectory, climbing has distinct trajectories for stance and flight, with the flight trajectory ending if stance is detected (when the position error in direction of the wall exceeds a hand tuned threshold). When stance is detected, the initial point of the stance trajectory is set to the current leg position while the terminal point remains fixed. The final nominal desired trajectory for both is shown in Fig. 6. The nominal shape for the running behavior was designed to approach the teardrop shape implemented on Minitaur. The flight phase of the climbing controller was designed with a small upward extension vertically to improve detachment from the wall at the beginning of flight and a smooth approach into the wall that reaches into the wall until a maximum length is reached or until stance is detected.

### 9.2 Experimental Results.

In climbing tests, the nominal 5-bar linkage morphology was incapable of climbing even with the constrained body dynamics. Motor effort for an average stride during a climb is shown in Fig. 7. The innermost motors of the nominal leg stalled mid-stance during both low and high driving frequencies, putting significant strain on the robots electronics and preventing completion of the stance trajectory. This stall occurred shortly after stance began in a similar kinematic region to what was predicted by the EDW. Using the near-optimal leg morphology the robot was able to successfully climb unsupported at up to 3 Hz. It was found that the innermost motor peaked at 49.4% of the full motor effort during climbing. During stance, the average torque distribution between the motors is 56.99%. This compares well with the simulation where the average work ratio for this linkage is 57%. Driving frequencies are limited to 3 Hz due to attachment issues with the substrate and inaccuracy of trajectory tracking without altering the stiffness of the virtual leg springs. The average climbing velocity at is approximately 17.5 cm/s at this frequency.

BOBCAT showed comparable performance when running (see Table 2) using both linkage sets. Both linkages were capable of traveling over 2 m/s at driving frequencies from 5 to 7 Hz. The peak velocity achieved was 2.3 m/s using the near-optimal linkage at 7 Hz. The computed velocities using the nominal leg were slightly below that of the new near-optimal leg at all driving frequencies tested.

The experimental results show that the DKLD was effective in producing a set of kinematic parameters for the coaxial 5-bar leg morphology which are capable of producing dynamic climbing. The EDW was shown to be a useful visualization tool as the motor of the original leg was found to experimentally stall near the region of the legs workspace predicted in Fig. 5. With only a single set of target dynamics used for the DKLD, a new leg design was found which allowed BOBCAT to achieve running and climbing multimodality.

## 10 Case Study 2: LLAMA Efficiency

One of the primary benefits of utilizing legged locomotion is the ability to handle rough terrain. When navigating unstructured environments, flexibility in locomotive behaviors, and robust sensing are highly valued. LLAMA is a large quadrupedal robot developed to carry a heavy, but sophisticated, sensor suite and body design as well as implement a diverse range of locomotive behaviors so that it may be deployed in real-world environments. The primary limitations currently experienced include power and efficiency. This leg design study’s goal is to evaluate whether a modified leg design can permit this robot to efficiently employ a diverse collection of behaviors which may be used in navigation, including gaits that enable crouching under obstacles and running at up to 2 m/s.

### 10.1 LLAMA Platform and Multiple Dynamics.

The LLAMA platform, shown in Fig. 8, is 60 kg robot that stands at up to a half a meter tall which was being developed by JPL at the time of this writing. LLAMA has legs with three actuated degrees of freedom with a rotational shoulder joint connected to a coaxial 5-bar linkage leg design with a long end-effector extending off of one of the secondary linkages from the pin joint connection between the primary and secondary linkage. This gives the leg the appearance of planar, dual-revolute (RR), serial leg using a four-bar mechanism for the power transmission to the second revolute joint. The motors used on LLAMA are custom made and make up 58.3% of the robot’s weight.

## 11 LLAMA: Bipedal SLIP

### 11.1 Model.

*M*is defined as half the mass of the robot (which is assumed to be using a trotting gait) and is being acted on by gravity with acceleration

*g*. Linear stiffness and damping of a single leg are noted by

*k*and

*b*, respectively, while the torsional stiffness and damping are noted by

*k*

_{R}and

*b*

_{R}. The force generated linearly in the leg is shown in Eq. (13) and the torsional force of a leg is shown in Eq. (14)

*L*and

*θ*and the actual linear and angular velocities are $L\u02d9$ and $\theta \u02d9$. The desired positions and velocities prescribed by the actuators are represented by the subscript

*des*. The equations of motion of the leg are shown in Eqs. (15) and (16)

*C*

_{1}and

*C*

_{2}are representative of a Heaviside function indicating contact. They hold the value of 1 when a leg is in contact with the ground and 0 when the leg is in flight. Flight phases are determined to occur when the vertical ground reaction force goes to zero. In double stance, constraint equations are applied to the equations of motion maintaining the fixed distance separating the feet at the initial ground contact

*L*

_{2}and

*θ*

_{2}(the length and angle of the leading leg on the ground) in terms of the length (

*L*

_{1}) and angle (

*θ*

_{1}) of the trailing leg on the ground as well as the fixed horizontal distance between the legs (

*L*

_{sep})

### 11.2 Behaviors and Control.

In order for LLAMA to leverage the benefits of using legs to flexibly navigate complex real-world environments, several behaviors were desired for locomotion. A set of three behaviors are chosen which represent walking in various parts of the workspace: these include a mid-workspace, crouch, and tall walk. The mid-workspace walk is meant to be representative of a standard walking mode while the crouch and tall walk are specialized gaits for obstacle navigation under and above obstacles, respectively. In addition to these, two running behaviors are constructed which will verify that the robot can achieve higher speed gaits with flight phases. Further descriptions of these gaits are discussed in the LLAMA Simulation section.

The leg was actuated using a feed-forward version of the active energy removal (AER) controller (Fig. 10(a)) described in Ref. [43]. AER has been shown to produce robust running behaviors to perturbation and parameter variations making it a good candidate for constructing the target dynamics for a robot while it was in development. Feed-forward control was selected as it allows the user to directly set actuation targets for a large area of the workspace and it lends itself well to systems with minimal sensory input.

Linear and angular actuation of a leg are phase shifted by $90\u2218$ sinusoids which oscillate around a nominal position (*L*_{nom} and *θ*_{nom}). The nominal positions and actuation magnitudes of these waves are selected based on the desired set of target dynamics being produced. The commanded actuation of the two legs is separated with a phase shift of $180\u2218$ with the flight leg continuing the feed-forward trajectory to determine the initiation of its stance period. In simulation, the model starts in single stance in which an initial phase shift *ψ*_{init} determines the actuation state at the beginning of stance. This parameter is included with the model’s initial conditions and tuned with during the Newton–Raphson search for fixed point. A sample of the foot trajectory formed by through these control inputs in operational space is displayed in Fig. 10(b).

## 12 LLAMA: Kinematics

LLAMA’s leg, shown in Fig. 10(c), is made up of a specific 5-bar kinematic chain which imitates biological legs (a serial, RR leg) while maintaining the actuators close to the center of mass. Links *L*_{1} and *L*_{5} each connect directly to one of the leg actuators. Links *L*_{3} and *L*_{2} make up a single rigid body with *L*_{2} deflecting from *L*_{3} at a fixed angle *α* initiating at the pin joint connection with *L*_{1}. The actuation of the *L*_{5}, *L*_{4}, *L*_{3} kinematic chain is meant to emulate the actuation of the knee in an RR leg.

*θ*

_{1}the angle between

*L*

_{1}and

*L*

_{5}, and

*θ*

_{2}the angle between the linear extension of

*L*

_{1}and

*L*

_{2}. The angle

*θ*

_{1}can easily be constructed from the difference in the motor angles

*M*

_{1}(connected to

*L*

_{1}) and

*M*

_{2}(connected to

*L*

_{5}). The angle

*θ*

_{2}can be derived using a combination of the laws of sines and cosines [44] and is shown in Eq. (23)

*L*

_{a}and for the inputs to the quadratic used to compute

*θ*

_{2}are shown in Eqs. 21 and 22, respectively. Using the motor angle

*M*

_{1}and

*θ*

_{2}the foot relative to the hip positions can be computed. The foot positions of relative to the hip are computed in cylindrical coordinates, where

*R*is the length of the virtual leg (shown in Eq. (24)) and the angle of the virtual leg (shown in Eq. (25))

*θ*

_{1}is constructed from the difference between the motor angles its partial derivatives

*δθ*

_{1}/

*δM*

_{1}and

*δθ*

_{1}/

*δM*

_{2}are −1 and 1, respectively. The differential of

*θ*

_{2}can be derived with respect to

*θ*

_{1}(Eq. (28)) and by propagating the motor specific partials

*θ*

_{2}is constructed from individual differentials of the equations used to compose quadratic input to the arc-tangent function used in Eq. (23). These are shown in Eq. (26). For the propose of clarity the additional interim Eq. (27) is used to formulate Eq. (28). Lastly, the Jacobian for the cylindrical coordinate system of the leg can be produced by differentiating Eqs. (24) and (25) with respect to

*θ*

_{2}

## 13 LLAMA Constraints

The primary constraint used on LLAMA, like BOBCAT, is the motor model. The LLAMA motors have a stall torque *T*_{Stall} = 21.9 N m and a no load speed of *ω*_{NoLoad} = 997.5 rad/s with a gear ratio of 5.25 as per specifications received from JPL. The dynamic model is subjected to limit cycle stability requirements which are used to determine fixed points for gaited locomotion.

The boundaries of the six-dimensional linkage space used in optimization are shown in Table 3. Links *L*_{1} and *L*_{2} (being the primary linkages to determine standing height) have bounds which contain the longest linkage lengths and span the largest ranges of length. The links of the power transmitting 4-bar (*L*_{3} to *L*_{5}) have shorter maximum and minimum linkage lengths in order to encourage a design which appears more bio-mimetic. Link *L*_{4} has the largest permissible variation amongst these as it is opposite *L*_{1} in the kinematic chain and may require longer lengths to close the chain. The upper bound of *L*_{5} is set to the maximum allowable length it can extend (Limited by the motor radius) without causing interference with the body. During optimization any combination of these linkages which cannot fully contain the motion profile of the target dynamics is given a high cost.

An additional constraint is placed on the LLAMA legs due to the robot’s opposing leg direction. As the front and rear legs each have the knee directed outward from the body, the target dynamics are applied by running them through the leg kinematics both forward and backward. This is achieved by mirroring the horizontal motion and force profiles about the legs’ center of mass. The legs performance, used for optimization, is constructed with the maximum torques from either leg direction and their respective angular velocities.

## 14 LLAMA Simulation

As in the previous study, the simulation for this design is performed in matlab. Parameters used to define the dynamic model can be found in Table 4. Stiffness and damping values were generated from a previously successful bipedal SLIP model [45] using dynamic scaling laws [46] to adapt them for LLAMA’s height and body mass. The nominal leg lengths *L*_{nom} are representative of the center of the sinusoidal extension actuation of AER with values based on the workspace size of the current LLAMA leg. The crouch walk uses the lowest value of 18.75 cm (centered around the lower 25% of the workspace), both the high step and running use 36.25 cm (centered upper 75% of the workspace), and mid-workspace walking uses the center of the workspace at 27.5 cm.

Walking gaits were actuated at 1.5 Hz with commanded extensions of 10% of the current workspace or 3 cm. The middle and tall walks had angular actuation spanning $30\u2218$, which generates swift walking for the middle walk and large angular foot steps in the tall walk. The crouch gait spanned only $10\u2218$ to produce smaller steps with the reduced vertical center of mass variance (desirable for carefully passing under an obstacle). Two running gaits were used here one found at 1.5 Hz (representing run with high vertical energy) and one at 3.5 Hz (representing run with high horizontal energy).

The variables *W*_{1} and *W*_{2} refer to the average work done by motor 1 (connected to *L*_{1}) and motor 2 (connected to *L*_{5}), respectively. Weighting of the work ratio is represented by the second expression in Eq. (31) which varies the weighting of power between 1 and 1.5 centering the value around an evenly divided work ratio of 50%. Efficiency was chosen to minimize the requisite power demand of locomotion on the robot’s battery, while the work ratio weighting reduces the uneven wear on the motors due to the load being concentrated on a single actuator.

The total cost of the leg design fed into the optimization algorithm, Eq. (1), is made up of two mode costs: a walking mode cost (composed of the norm of the walk, crouch, and high step behaviors) with a mode running cost (composed of the norm of the running gaits). The walking and running are specified as separate costs to encourage the optimizer to favor legs which allow both distinct locomotive modes.

### 14.1 Results.

The body positions and ground reaction force profiles of the locomotive behaviors used to construct the target dynamics on LLAMA are shown in Fig. 11. Although behaviorally all of the walking gaits experience similar profiles (i.e., double peaks in vertical ground reaction force and horizontal braking at the onset of double stance), each gait experiences forces at different magnitudes and operate in different locations in the workspace. Each gait, therefore, produces a unique set of requirements for design. Similarly, in the two running gaits, it can be seen that the locomotion imposed distinctly different force magnitudes both representative of the running style. The 1.5 Hz run imposes much higher vertical forces while the 3.5 Hz run requires larger lateral forces.

When trying to produce these target dynamics on initial LLAMA leg design, both of the running behaviors fail due to the requisite torque exceeding the motor limits. When observing the weighted efficiency costs of the remaining walking gaits, the crouch has a cost of 85, the mid walk a cost of 119, and the tall walk a cost of 216. In simulation producing the tall walk would require peaks of 95% of the available motor power and for the 1.5 Hz run the motor would require nearly double the available torque. The work ratio in these gaits averages around 60%, allocating the load more to the motor attached to *L*_{1}.

The optimization produced a leg with lengths *L*_{1} = 15 cm, *L*_{2} = 30.2 cm, *L*_{3} = 13 cm, *L*_{4} = 21.7 cm, *L*_{5} = 6.3 cm, and a joint angle of $\alpha =20\u2218$. Across ten repeated optimizations the results converged in fewer than 170 iterations and showed less than 1.17% variation in kinematic parameters. This leg can perform all of the prescribed behaviors composing target dynamics while reducing the weighted efficiency cost of the crouch walk by 40% and the mid walk by 20%. The cost of the tall walk remains nearly the same however the requisite torque is reduced to 80% of stall. The running costs are significantly higher than that of the walking gaits (due to higher total power requirements) however the optimization found a set of kinematics which could produce both of these behaviors within the constraints.

A comparison is made between the EDW’s of the initially designed LLAMA leg and the optimized LLAMA leg, shown in Fig. 12. This EDW is formed using simplified, horizontal and vertical leg trajectories which enforce a conservative approximation of the forces and velocities (derived from the running behaviors) needed in locomotion within the localized regions of the workspace. The optimized leg design increases the minimum leg length by 2 cm and decreases the maximum leg length by 5 cm, however, it improves the usable workspace by approximately 130% for horizontal behaviors and 260% for vertical behaviors. The initial design performs poorly in the middle of the effective workspace used for walking and running. The usable region of this leg relies on its proximity to the kinematic singularities of the leg, meaning behaviors which require large vertical lengths changes (such as stair climbing) would be difficult with this morphology. Although the total workspace shrinks, the optimized leg performs much better in the middle of the effective workspace facilitating more behavioral diversity on the robot.

The optimized design shows significant improvements over the original design, enhancing both the capabilities and performance. Although simulation showed that the original leg is technically capable of producing the three walking gaits, both DLKD and the EDW show that this leg approaches actuator limits in these regions and may be pushed to stall in the presence un-modeled dynamic forces. The optimal leg, however, successfully completes all behaviors even under the conservative estimates provided in the EDW. It also does this while simultaneously improving the average efficiency of the walking gaits by more than 40%. The work shown here demonstrates the theoretical optimal leg for LLAMA under this objective and constraints. The chosen leg used on the LLAMA robot [47] was also designed using the DKLD method using slightly different objectives and constraints.

## 15 LLAMA Multibody Simulation

To further study the optimized leg design, the Simscape multibody physics engine was used. The completed quadrupedal LLAMA CAD model (Fig. 8) was imported into the Simscape environment with the individual component masses set based on the geometry and material properties. The legs designs are characterized by the primary kinematic parameters and therefore maintain similar shape properties when the legs change size with some moderate changes to linkage mass. This modeling approach for the LLAMA robots follows the approach used in Ref. [47].

### 15.1 Simulation Design.

*K*

_{ground}= 10

^{6}N/m and ground damping

*B*

_{ground}= 10

^{3}Ns/m. A Coulomb friction model is used with static friction

*F*

_{ST}= 1 and kinetic friction

*F*

_{KE}= 0.7. These contact models were selected based on previous modeling studies [50]. To constrain the actuator capabilities, a DC motor model was used on each actuator, with actuator torque

*τ*limited by

*ω*is the angular velocity,

*T*

_{Stall}= 26.667 Nm is the stall torque, and $WNL=997.5RPM$ is the no load speed. In addition, all simulation runs were constrained by the electrical power limitations of the physical robot. Electrical input power

*P*for each actuator is given by

*R*

_{m}= 10.8 Ω and the torque constant is

*K*

_{t}= 1.95. The gear ratio for the motors is

*GR*= 5.25. Table 5 summarizes the parameters in the contact model and motor models used within Simscape for the quadruped.

### 15.2 Controller.

Each $3deg$ of freedom leg is defined as a revolute, revolute, prismatic (RRP) manipulator with generalized coordinates *R*, *ϕ*, and *ψ*. Using this virtual manipulator a PD controller is defined on the three generalized coordinates, which allows us to effectively specify stiffness and damping terms in each dimension.

Locomotion of this model is produced using a feed-forward (Buehler clock based) triangular trajectory similar to controllers which were initially employed on Minitaur [40]. The leg coordination followed a trotting gait, which commanded a duty factor *DF* and a 50% phase shift between diagonal leg pairs based on a fixed cycle frequency *f*.

The shape of the trajectory is defined by the stroke length *L*_{stroke}, nominal leg length *L*_{nom}, flight offset *L*_{off}. *L*_{stroke} defines the forward distance for each foot to travel, while *L*_{off} provides an adjusted apex location of the 3-point trajectory to assist with maintaining the forward velocity of the robot. These control parameters can be seen in Table 6.

### 15.3 Results.

In the above equation *P* is the power consumption of the robot, *Mg* is the robot weight, and $v\u2192$ is the average fore-aft velocity of the body during steady-state walking. These results can be seen in Table 7. The optimal LLAMA leg was able to achieve a 45% reduction of cost of transport from the original leg design, with the optimized leg’s *COT* = 0.88 at steady state. The optimized leg even achieves a higher steady velocity, with $v\u2192=1.905m/s$ for the original leg design and $v\u2192=2.074m/s$ for the optimized leg design.

## 16 Case Study 3: Small Multimodal Platform

The final study considers multiple dynamics and multiple leg mechanisms for the development of a new multimodal quadrupedal robot, at half the size of BOBCAT. Due to BOBCAT’s size and weight, very few real-world climbing substrates could withstand the reaction forces generated by the robot during climbing. This issue is particularly problematic when attempting to achieve high-speed transitions between running and climbing modes. As such, development has begun on a smaller multimodal quadruped, here after referred to as Mini-BOBCAT.

Mini-BOBCAT is projected to be half the size of the current BOBCAT platform. It uses two dc motors per leg (Maxon DCX 14L 14D, 6V motor) producing two actuated degrees of freedom in the sagittal plane. The leg design will take the form of a power transition mechanism (either a parallel linkage or chain drive) that allows the actuators to be centralized on the body. Given these design parameters this study endeavors to determine the optimal leg design for this robot.

## 17 Mini-BOBCAT Models

This newly designed robot is expected to achieve similar locomotive capabilities of other quadrupedal running robots while also being capable of multimodality via dynamic climbing which has distinct dynamic requirements. Therefore the target dynamics in this study are constructed using both the modified FG model and the linearly and torsionally actuated bipedal SLIP model described previously. The control of these models mimics that of the previous studies i.e. fixed velocity linear retraction in climbing and a feed-forward AER control for ground traversal. As on LLAMA, a set of walking and running behaviors are developed using the bipedal model. As before, walking behaviors are composed of a crouch, mid length, and tall walk based a workspace at half scale from the original BOBCAT leg. Two running behaviors are also produced, one held at mid leg length and one at the upper leg length. Beyond walking and running an additional behavior was developed in the horizontal domain: a generalized jumping behavior.

### 17.1 Pounce.

The jumping behavior added to the target dynamics of Mini-BOBCAT, referred to as the pounce, is shown in Fig. 13. This is a generalized jumping behavior meant to enable the freedom to jump over and on top of obstacles, as well ensuring the robot has the framework for a high-speed transition from running to climbing modes, to be developed later. This pounce maneuver is characterized as a counter-jump in which the leg is initially retracted and then extended at high speed all the while rotating toward a target liftoff angle.

The dynamics for the pouncing behavior are generated using the linearly and torsionally actuated bipedal SLIP model. The legs of the robot act in phase with one another, initially in double support using the same contact point with the ground (effectively becoming a single leg acting with parallel stiffness and damping values). The leg is actuated using the previously described feed-forward AER controller with the extension frequency acting at twice the rate of the angular sinusoid. As the initial controller phasing begins with the leg fully extended and vertical, the doubled actuation rate produces the counter-jump behavior.

The pounce is an occasional behavior used for obstacle negotiation and therefore does not need continuous stability requirements as a success criterion (as in the SLIP or FG models). A success criterion for this behavior was set to be the leg lifting off within $5\u2218$ of the desired lift off angle. In this study, 3 Pounce behaviors were used, setting lift of angles at $45\u2218$ (a distance jump), $75\u2218$ (an obstacle-clearing jump), and $90\u2218$ (a vertical jump).

## 18 Mini-BOBCAT Kinematics

Unlike the previous studies, this robot is still in its earliest development and the legs are not morphologically restricted. Therefore three distinct sets of leg kinematics that make use of the two actuated degrees of freedom are used to explore possible leg designs for this robot. These include: the coaxial 5-bar used on BOBCAT (BC), the LLAMA 5-bar (LL), and a planar revolute-revolute (RR) serial leg similar to that of the planar representation of shoulder and knee joints used by many quadrupedal animals and terrestrial robots [15,51].

### 18.1 Planar Serial RR.

The kinematic equations for a serial RR are well defined for manipulators and are easily constructed using the Denavit-Hartenberg conventions [52] and therefore not explicitly shown here. During running the knee joint connection between femur link (*L*_{1}) and tibia link (*L*_{2}) is constrained to be behind the virtual leg length *R*. Similarly in climbing the knee joint is held toward the wall assuming a linear toe extension along *L*_{2} to determine the approach angle to the substrate.

## 19 Mini-BOBCAT Constraints

The linear motor model is used in simulation for the Maxon DCX 14L 14D, 6 V motor which has a stall torque of *T*_{Stall} = 0.248 Nm and a no load speed of *ω*_{NoLoad} = 1090 rad/s. This actuator is considered to be run at 2.47 times the nominal voltage (as if powered by 14.8 V lithium polymer battery). A 35:1 planetary gear head is used to have sufficient speed and torque to locomote based on a preliminary design point which approximated a speed and torque requirement.

Linkage boundaries for Mini-BOBCAT’s leg optimization are shown in Table 8. Optimization boundaries for the primary linkages (motor connected links: *L*_{1} in all legs, *L*_{3} in the BOBCAT 5-bar, and *L*_{5} in LLAMA leg) are set based on the projected frame design. The lower boundary is set to a length one centimeter greater than the motor radius and the upper bound is set to be a centimeter shorter than the hip bracket in order to allow for sufficient clearance for the construction of the pin joints between links. Link *L*_{1} in the LLAMA leg is allowed a slightly longer upper bound as the extension of *L*_{2} already prevents the rotation of this link through the bracket. The secondary linkage lengths (*L*_{2} in all configurations and *L*_{4} in the BOBCAT leg) are set to allow approximately the same stand height scaled at half the BOBCAT’s leg length. The power transmitting linkage ranges in the LLAMA 5-bar are scaled to the size of Mini-BOBCAT with the angle *α* allowing a larger range to increase the number of configurations which have appropriate attachment angles for climbing.

Climbing constraints take a similar form to that in the BOBCAT study: no elbow or body contact with the wall, zero pitch-back moment during locomotion, and a limited range of attachable angles. In order to ensure no contact with the wall outside of the foot the length *D*_{body} in the out-of-plane model (Fig. 3(b)) is always given a half centimeter of clearance with the elbow or body (whichever is the longer length). For BOBCAT the elbow is the joint between *L*_{3} and *L*_{4} where as in the serial RR leg this is the joint connecting *L*_{1} and *L*_{2}. Due to the knee joint extension in the LLAMA 5-bar (the *L*_{3} to *L*_{4} connection) having a highly likelihood of both wall and body interference the forelimbs hold this joint in front of the body for running and therefore above the body during climbing. This also means that, like the LLAMA study, the LLAMA kinematics is simultaneously optimized for ground traversal in both directions. The positioning of the climbing trajectory in the workspace for the LLAMA leg is chosen to be at the body clearance length in order to minimize the out-of-plane forces. As in BOBCAT this distance is fixed, with regulatory forced computed based on the quasi-static out-of-plane model for zero-induced pitch. The allowable attachment range of attachment angles is again set to be between $+15\u2218$ and $\u221210\u2218$ from parallel with the wall. The target dynamics of climbing for this robot set a minimum stroke length of 7 cm to allow the leg ample time to accurately reset in flight at high speed.

## 20 Mini-BOBCAT Simulation

Nine separate behaviors were constructed to optimize the design of Mini-BOBCAT. The physical parameters of the dynamic models are shown in Table 9. The body mass of the robot is set to be half of the projected body mass as both running and climbing are set to perform trotting gaits which shares the load between diagonal leg pairs. The pounce behavior uses stiffness and damping values equal to those of the bipedal running model with the legs acting in concert, as the jumping behaviors are performed with all four legs of the quadruped.

Walking behaviors (the crouch, mid, and tall walks) were actuated at a cycle frequency of 2.5 Hz with extension magnitudes of 15% of the projected workspace (based on a half scale Minitaur workspace). The mid run was set at 5 Hz with a 15% extension and the tall run at 4 Hz with a 20% extension in order to encourage optimization to select legs capable of performing a range of running behaviors. The linear actuation of the pounces proceeds at 5 Hz (corresponding to a 2.5 Hz angular actuation) with commanded extension through 50% of the nominal workspace. A sweep of climbing behaviors was produced varying the nominal leg length (i.e., the center of the 7 cm retraction). This was chosen to increase the number of possible legs which can perform the climbing behavior within the rigid foot positioning and attachment angle constraints. As on BOBCAT, during optimization a single climbing behavior is selected from this sweep which chooses the *L*_{nom} value closest to the center of the effective climbing workspace.

Optimization for the entire set of target dynamics was performed for each set of leg kinematics using both the peak torque minimizing and work weighted efficiency cost functions. The resulting optima are presented below.

### 20.1 Results.

The optimization successfully found six optimized leg designs, one for each combination of leg kinematics and cost function. The subcosts for each behavior of the optimized legs can be found in Table 10. For the RR leg, the torque minimized optimum leg converged to *L*_{1} = 4.43 cm and an *L*_{2} = 10.81 cm (with a much shorter primary linkage than secondary link) while the efficiency optimization converged to *L*_{1} = 7.15 cm and an *L*_{2} = 9.00 cm (closer to symmetry reaching to the lower boundary on the secondary link). Both optimizations converged in less than 206 iterations and $2.4*10\u22127%$ variation in linkage lengths upon ten repeated optimizations.

For the coaxial 5-bar leg used on BOBCAT (BC), the torque minimizing cost function found a morphology with linkage lengths of *L*_{1} = 4.89 cm, *L*_{2} = 11.24 cm, *L*_{3} = 5.56 cm, and *L*_{4} = 10.95 cm. This configuration similarly favored the slight asymmetry found in climbing on BOBCAT, with *L*_{3} > *L*_{1} and *L*_{2} > *L*_{4}. The efficiency optimization converged to a 5-bar linkage with lengths of *L*_{1} = 5.05 cm, *L*_{2} = 9.23 cm, *L*_{3} = 5.25 cm, and *L*_{4} = 9.75 cm, which showed similar behavior in the primary linkages an opposing behavior in secondary links with *L*_{4} > *L*_{2}. Here optimizations converged in less than 260 iterations and 1.3% variation in linkage lengths upon 10 repeated optimizations. As in the RR, both of these legs were able to complete all of the desired locomotive behaviors.

The torque optimized LLAMA (LL) leg converged to *L*_{1} = 3.90 cm, *L*_{2} = 9.78 cm, *L*_{3} = 7.50 cm, *L*_{4} = 5.49 cm, *L*_{5} = 3.50 cm, and $\alpha =\u221257.7\u2218$, and the efficiency optimal leg converged to *L*_{1} = 5.26 cm, *L*_{2} = 9.06 cm, *L*_{3} = 6.40 cm, *L*_{4} = 4.93 cm, *L*_{5} = 3.88 cm, and $\alpha =\u221259.3\u2218$. Both optimal legs approached the lower boundary on parameters *L*_{5} and *α* and near the upper boundary of *L*_{3}. Optimization was unable to find a LLAMA leg which could perform the climbing behavior, due to the leg being incapable to meet the prescribed incident angle constraints and the motor stall constraint simultaneously. These results converged in less than 160 iterations and 15% variation in linkage lengths upon ten repeated trials. The large parameter variations and lower iteration count appear to be a result of the failure to climb.

## 21 Discussion

### 21.1 Subcosts.

As seen on the resulting optimization subcosts (Table 10) the optimum RR legs had the highest average costs under both cost functions. This leg, comparatively, performed best in the pouncing gaits due, in part, to significant commanded extension driving this behavioral set. Using this leg better distributes the requisite power between the actuators when extending from the hip in the workspace region where these are defined.

The coaxial 5-bar BOBCAT leg outperformed the RR leg in most categories except for the angled pouncing behaviors, in the efficiency optimized leg and the Tall walk and $45\u2218$ pounce, in the torque optimized leg. Significant improvements are seen in both the climbing and running behaviors which apply much larger forces in the lateral direction.

Outside of the failure of the robot to climb, the LLAMA 5-bar had the lowest costs of any leg. These improved costs arise due to two primary factors: a higher-dimensional linkage space in which to search and the competing costs of the climbing objective effectively dropping out of the optimization. As no legs were able to successfully climb, either due to motor or touchdown angle constraints, the high failure cost acted more as an offset to the cumulative cost, effectively optimizing a smaller set of target dynamics. These legs’ highest relative costs are in the pouncing behaviors. As this is similar to the results found in the LLAMA case study, this implies the LLAMA 5-bar morphology has difficulty performing highly energetic vertical behaviors.

### 21.2 Effective Dynamic Workspace.

To better visualize the designs’ performances, an EDW plot is produced for each of the resultant optima, shown in Fig. 14 for the six different designs (A–F), using simplified horizontal and vertical motion profiles as in the previous studies. Both peak torque and work ratio are presented for each leg design and motion profile. For the RR legs (A and D), peak torque in vertical motions depends highly on knee placement and performed poorly relative to the other legs. When optimizing for peak torque (A) it can be seen that the leg selected generally reduces the torque magnitude at the cost of the leg’s workspace. This set of kinematics also has large changes in work distribution, frequently placing greater stress on either motor. This is particularly poor for climbing regions in which almost all of the work is done by a single motor.

The legs using coaxial 5-bar linkages (BOBCAT and LLAMA) demonstrate workspaces with generally reduced peak torque magnitudes, particularly for primarily vertical motions. However, due to more regions near kinematic singularities using these simplified behaviors they show small “tears” of stall in Fig. 14. Similar to the RR configuration the LLAMA 5-bar demonstrates significant work ratio variance with a slightly more balanced distribution in the work-weighted efficiency optimal leg (F). The BOBCAT 5-bar, however, has a much more consistent work distribution, most noticeably in the vertical direction averaging to less than a 5% change from evenly apportioned. The improved distribution is particularly advantageous for climbing as this allows for a significant reduction in the peak torque for BOBCAT legs compared with the RR.

### 21.3 Design Selection.

Any of the RR or the coaxial BOBCAT 5-bar legs are viable designs for Mini-BOBCAT. The LLAMA kinematics are not feasible as they are unable to climb without making special design considerations to facilitate attachment at poor angles. When considering the design goals for implementing this on a multimodal platform, the legs that stand out are the efficiency optimal RR leg (D) and the peak torque optimal BOBCAT 5-bar (B).

The efficiency optimal RR leg, although having the highest torque requirements for vertical behaviors, has the largest workspace of any of the viable legs. Given that the minimum requisite dynamic behaviors are achievable in locomotion, the additional space provided makes other tasks (such as manipulation of objects in the environment) more accessible. Additionally, the simplicity of the control for this set of kinematics facilitates faster computation and enables processing power to be allocated elsewhere.

The peak torque optimal BOBCAT 5-bar not only has the benefits of significantly reduced running and climbing stresses and even work distribution but also has some reliability as this identical design process has already been shown to be effective on BOBCAT. The torque abatement in this design is valuable for managing any un-modeled forces which may be incurred during locomotion, which would be particularly problematic for climbing using the optimal RR leg. Both of these legs will be tested as possible options for the design of Mini-BOBCAT.

## 22 Conclusion

This work presents a novel approach for analyzing the suitability of leg designs called “dyno-kinematic leg design” and a workspace visualization tool called the “effective dynamic workspace.” DKLD utilizes known or computable locomotion dynamics as targets in the process of optimizing a robot’s leg kinematic parameters to produce high performance power transmissions, while EDW provides a way of clearly seeing the design differences. These techniques are demonstrated in three separate case studies: enabling dynamic climbing on BOBCAT, improving efficiency on LLAMA, and selecting a leg design for a new, small, multimodal quadruped.

In order to capture the desired target dynamics on BOBCAT a new instantiation of the pendular dynamic climbing model is used. These forces and velocities are mapped through a set of coaxial 5-bar kinematics to determine the viability of a specific set of linkage lengths which were subsequently optimized. The resulting leg shrunk the effective workspace by 32% from the nominal configuration, however, the EDW of the leg more than doubled in size. Using new near-optimal linkages, the robots were able to reduce its peak torque in climbing from a complete stall to less than half of the motors max effort while maintaining running velocities over 2 m/s. The design technique successfully enabled BOBCAT to dynamically climb vertical substrates, with additional experiments showing it could even do so bipedally.

In order to improve the efficiency of the large-scale LLAMA quadruped several desirable locomotive behaviors were generated using a linearly and torsionally actuated bipedal slip model. Behaviors were selected to act in a broad range of the nominal legs’ workspace and consisted of both walking and running gaits. The target dynamics were mapped from foot to motor space by constructing the kinematics and jacobian for the LLAMA 5-bar leg. Optimization of this set of kinematics produced a leg which improved the workspace size for high stress horizontal motion by 130% and for vertical motion by 260%. These linkages were verified to improve the robot’s efficiency in a multibody simulation.

The third study, the multimodal robot Mini-BOBCAT, utilized nine different locomotive behaviors to make up its set of target dynamics and three sets of linkage kinematics. The dynamics included a climbing behavior (captured using the modified Full-Goldman climbing model), a set of walking and running behaviors, and a set of pouncing behaviors (captured using the linearly and torsionally actuated bipedal slip model). The three linkage kinematics, each with two degrees of actuated freedom, explored for leg design were a dual-revolute leg, the BOBCAT 5-bar, and the LLAMA 5-bar. These legs were optimized under two separate cost functions producing six different optimal legs which each performing slightly differently due to the nature of their respective kinematics. From this, a pair of viable designs are selected for future implementation and testing after the robot hardware is developed.

The three case studies have shown that DKLD can reduce peak torque, improve workspace efficiency, and satisfy a range of potentially conflicting target behaviors. It differs from mechanism-centric and bio-mimetic design in that it directly incorporates the dynamics of locomotion into its design procedure. Notably, although some similarities can be seen in the trends of certain kinematic parameters (which help produce a more favorable mechanical advantage), each tested leg morphology converged to distinctly different values based on the tasks, objectives, and constraints. Suggesting that what is optimal is highly dependent on how the designer wants to achieve the behaviors which DKLD allows for a lot of freedom in that respect.

This method does not replace control design, but rather supports and can be adapted to it. The DKLD procedure can provide a more capable plant, increase the available actuator authority and simplify the control problem. A high performance controller (which may be optimally designed or informed by sensory data) could also be used as an input to the target dynamics and thus be used to generate legs which best execute the desired robot behaviors. The designer’s ability to input such locomotive insights makes DKLD a powerful tool to integrate difficult and diverse mechanism designs, motions, and objectives into a centralized approach.

## 23 Future Work and Further Application

Though the most immediate extension of this work is the application of the newly designed legs from case study 3 into a new robot which is currently in development, this design technique has many possible avenues for future research and applications. As the concepts of DKLD and EDW involve the selection of both a set of target dynamics and a morphology, this technique may be extended to a multitude of legged robots. This technique may also be augmented through the fusion of target dynamics from multiple sources i.e. constructed from simulation data or perhaps experimentally collected biological or robotic data. Both the control and design have an impact on the performance and capabilities of a legged robot. the complex relationship between these suggests it may be beneficial to co-optimize the design and control. Optimized trajectories may be utilized as the target dynamics for the DKLD method. Sequential use re-optimizations of the control and kinematics may generate a high performance robot even with significant constraints.

Another expansion on this work can arise from leg specialization. In this paper, we assumed similar near identical roles for each leg on quadrupedal robots. However, this technique may easily be used to design role specialized legs for multilegged systems. If different target dynamics can be attained for each leg on a 4-legged, 6-legged, or 8-legged robot, DKLD could be applied to optimize each leg independently and produce a set of morphologically distinct legs with different workspaces while still guaranteeing the necessary locomotion is encoded into each legs design. In a similar vein, this may be used to study the design motivations for animals. Many animals with more than two legs (such as dogs [53]) have slightly different designs and functions for each fore-aft leg pair, which have evolved to better suit their purpose. By using biological data and analyzing optimal leg designs we may be able to study the nature of evolutionary objective functions for different animals.

## Footnote

Preliminary results of this first case study have previously published in Ref. [24].

## Acknowledgment

This work was supported by the collaborative participation in the Robotics Consortium sponsored by the U.S. Army Research Laboratory under the Collaborative Technology Alliance Program, Cooperative Agreement DAAD 19-01-2-0012, and by NSF Grant CMMI-1351524. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes not withstanding any copyright notation thereon.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.