Legged Mechanism Design with Momentum Gains Brandon J DeHart and Dana Kuli´ c Abstract— There are two main goals for any mobile, bipedal system: locomotion and balance. These behaviors both require the biped to effectively move its center of mass (COM). In this work, we define an optimization framework which can be used to design a biped that maximizes its ability to move its COM, without having to define an associated controller or trajectory. We use angular momentum gain in our objective function, a measure of how efficiently a system can move its COM based on its physical properties. As a comparison, we also optimize the model using a cost of transport-based objective function over a set of trajectories and show that it provides similar results. However, the cost of transport calculation requires slow hybrid dynamics equations and hand-designed trajectories, whereas the angular momentum gain calculation requires only the joint space inertia matrix at each configuration of interest. I. I NTRODUCTION The two critical performance objectives of most bipedal systems are its ability to locomote and to balance. Initially these appear to be fundamentally different behaviors: • The goal of balancing is to keep the biped’s center of mass (COM) in a desired upright pose or on a trajectory, without undesired changes in the contact surface(s). • The goal of biped locomotion (gait) is to move the COM in space, via repeated changes in contact. However, both of these behaviors can be more easily accomplished when a biped is able to efficiently move its COM relative to the contact(s). When balancing, the COM is moved towards a point above the contact(s), while for gait it is moved between a series of contact points. This is true both in the static case, where the COM should be maintained above the contact surface(s), and the dynamic case, where the COM is moved between contact surfaces. In both of these cases, compensating for external disturbances also typically requires COM movement. Therefore, a biped which can efficiently move its COM relative to its contact(s) should be excellent at these behaviors. The ability of a biped to balance and walk is impacted by both the physical properties of the biped and the controller used to achieve the desired behavior(s). In this paper, we develop an optimization framework which can be used to design a biped with excellent balance and gait capabilities within a specified motion space, regardless of the controller used to achieve those behaviors. The objective function used in the optimization is based on the model’s angular momentum gain, a measure of *This work was supported by the Natural Sciences and Engineering Research Council of Canada. Brandon J DeHart and Dana Kuli´ c are with the Department of Electrical & Computer Engineering, University of Waterloo, Ontario, Canada. {bjdehart,dana.kulic}@uwaterloo.ca how effectively a mechanism can move its COM initially proposed in [1] for 2-link planar inverted pendulum models, and extended to general 2D and 3D models in [2]. Angular momentum gain is a measure of how efficiently an articulating system balancing on a passive (point or rolling) contact can move its COM via actuated joint motions. It is independent of the controller used, invariant to gravitational or velocity product dynamics, and (when balancing on a point or line contact) independent of the contact angle. In this paper, we use angular momentum gain to quantify and optimize the efficiency of COM movement based purely on the physical properties of a mechanism. The proposed approach is demonstrated on a 5-link biped mechanism. The rest of the paper is organized as follows: After summarizing the related work in Section II, we describe our optimization framework and our proposed objective function based on angular momentum gain in Section III. We demonstrate the capabilities of this framework using a 5-link planar biped in Section IV. We compare the results of our proposed objective function to a cost of transport-based objective in Section VI. Finally, conclusions and suggestions for future work can be found in Section VII. II. RELATED WORK Several research groups have used optimization to generate dynamic parameters for bipeds [3]–[9]. In these examples, the objective is to generate a gait for the biped in tandem with selecting its physical properties, using either the number of steps or the cost of transport as an optimization metric. In [3], [4], evolutionary computing and genetic algorithms were used to generate dynamic properties and control param- eters (or, somewhat equivalently, the trajectory) in tandem. These were the first examples of using optimization to generate the properties of biped mechanisms. More recently, a general framework was developed to extract design principles from biology [5], [6]. After obser- vation of a biological system, principles are transferred to a non-dimensionalized design space, which is then sampled and tested using an optimized controller to determine if the principle holds for the proposed design space. An optimization has also been developed to simultane- ously generate gait and design parameters in [7], although the only design parameter that is included in the optimization is a spring constant between the model’s thighs. A hybrid zero dynamics approach [10] is used to reduce the biped to a 1 DOF system, controlled with a trajectory tracking controller. In [8], a simulation framework was used to optimize the design and control of bipeds in parallel. The key differences between [8] and classic biped optimization methods such as