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Research Papers: Mechanisms and Robotics

# Design and Analysis of a Class of Planar Biped Robots Mechanically Coordinated by a Single Degree of Freedom

[+] Author and Article Information
J. McKendry

Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210mckendry.8@osu.edu

B. Brown

Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210brown.2225@osu.edu

E. R. Westervelt

Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210westervelt.4@osu.edu

J. P. Schmiedeler

Department of Mechanical Engineering, The Ohio State University, Columbus, OH 43210schmiedeler.2@osu.edu

Here, “dynamic gait” refers to a gait in which the biped’s foot-rotation indicator point (5) is outside of the support polygon.

Since the biped is planar, frontal plane stability is ensured by including a third leg whose motion is slaved to the motion of the other outer leg. Note that the inertial properties of the outer two legs combined must match those of the inner leg.

Justification for this assumption is presented in detail in Sec. 3.

The superscripts “−” and “+” are used from this point forward to denote values immediately before and after impact, respectively.

Since $qm$ is assumed to be monotonic, only a single value of $qi$ exists for each value of $qm$.

For the stability analysis, $qi$ is taken as the angle of the input relative to the absolute angle $qm$. Similarly, $hd(qm)$ is the desired relative evolution of the input angle $qi$.

Attempts to perform a complete kinematic optimization (femur and tibia simultaneously) were made, but the approximations of both the femur and tibia angles were less accurate with this approach. This may be attributed to the solution converging to local minima in the high-dimension, nonconvex optimization problem.

In terms of the mechanism input angle $qi$, the duration of each step is $π$ radians of rotation.

Maxon dc motor (Order No. 118752) and planetary gearhead (Order No. 166165) are selected.

J. Mech. Des 130(10), 102302 (Aug 21, 2008) (8 pages) doi:10.1115/1.2965609 History: Received June 26, 2007; Revised June 16, 2008; Published August 21, 2008

## Abstract

This paper presents a method of integrating mechanism design and hybrid system analysis for the design of single-degree-of-freedom (DOF) planar biped robots that can achieve dynamic walking gaits that are stable. Reducing the DOF in a biped can result in a reduction in the complexity of the control strategies needed to enable stable walking. Although the biped designed by this procedure is restricted to a single gait, this biped may be less complex, lighter, and less costly to construct than one whose multiple DOFs are coordinated via feedback.

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## Figures

Figure 1

3D rendering of a single-DOF planar biped. The biped’s design results in the need for only a single actuator. Here, frontal plane stability is ensured by including a third leg whose motion is slaved to the motion of the other outer leg.

Figure 2

Linkage configuration of the mechanism used in the example. Although the parameters describing both the stance and swing leg (qi, qm, and qb) are illustrated, only the stance leg mechanism is shown.

Figure 5

Joint angles and velocities versus time for a three-step simulation with the initial condition on the periodic orbit. Trajectories of qi are given as solid lines while trajectories of qm are given as dashed lines.

Figure 3

Mechanism schematics. The four-bar mechanism depicted in (a) is designed to approximate (only) the femur motion. The six-link mechanism depicted in (b) is designed to approximate the femur and tibia motions. The initial four-bar (femur) mechanism is unchanged (except for the input angle) when it is incorporated into the six-link mechanism. r⋆ is a vector that extends from the starting point p⋆s to the ending point p⋆e of each link. In (a) all angles are measured with respect to the positive x-axis.

Figure 4

Comparison of mechanism-generated (femur: dashed-dotted, tibia: dashed) and target motions (femur: solid, tibia: dotted)

Figure 6

Required actuator torque and absolute power versus time for a three-step simulation with the initial condition on the periodic orbit

Figure 7

Ground reaction forces versus time for a three-step simulation with the initial condition on the periodic orbit

Figure 8

Transmission angles for the four-bar femur and the five-bar tibia submechanisms for a one-step simulation with the initial condition on the periodic orbit. The solid lines correspond to the transmission angles of the stance leg mechanism while the dotted lines correspond to the swing leg mechanism. Note that the transmission angles of each leg (right leg and left leg) are continuous functions when considering the stance and swing leg roles that each leg assumes.

Figure 9

Joint angles and velocities versus time for a 20-step (only the first ten steps shown) simulation with the initial condition off the periodic orbit. Trajectories of qi are given as solid lines while trajectories of qm are given as dashed lines.

Figure 10

Slices of the state space for a 20-step simulation with the initial condition off the periodic orbit. The initial condition is indicated by an asterisk. (a) q̇i versus qi; (b) q̇m versus qm.

## Errata

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