Research Article: Arbitrary Symmetric Running Gait Generation for an Underactuated Biped Model

Date Published: January 24, 2017

Publisher: Public Library of Science

Author(s): Behnam Dadashzadeh, Mohammad Esmaeili, Chris Macnab, Johnny Padulo.


This paper investigates generating symmetric trajectories for an underactuated biped during the stance phase of running. We use a point mass biped (PMB) model for gait analysis that consists of a prismatic force actuator on a massless leg. The significance of this model is its ability to generate more general and versatile running gaits than the spring-loaded inverted pendulum (SLIP) model, making it more suitable as a template for real robots. The algorithm plans the necessary leg actuator force to cause the robot center of mass to undergo arbitrary trajectories in stance with any arbitrary attack angle and velocity angle. The necessary actuator forces follow from the inverse kinematics and dynamics. Then these calculated forces become the control input to the dynamic model. We compare various center-of-mass trajectories, including a circular arc and polynomials of the degrees 2, 4 and 6. The cost of transport and maximum leg force are calculated for various attack angles and velocity angles. The results show that choosing the velocity angle as small as possible is beneficial, but the angle of attack has an optimum value. We also find a new result: there exist biped running gaits with double-hump ground reaction force profiles which result in less maximum leg force than single-hump profiles.

Partial Text

The field of legged locomotion has proposed some fundamental models, including the Spring Loaded Inverted Pendulum (SLIP), the active SLIP, and the Point Mass Biped (PMB). These models allow one to investigate both walking and running gaits. The SLIP model remains a popular tool to investigate bipedal walking and running gaits ‎[1]. This passive model, consisting of a point mass and a massless spring leg, can generate a center-of-mass (COM) trajectory and a ground-reaction-force (GRF) profile similar to that of human running ‎[2]. This has encouraged researchers to investigate SLIP to try to understand the fundamental dynamics of bipedal running ‎[3], and also to use it as a template for controlling real biped robots ‎[4]. For any forward velocity, SLIP can generate periodic running gaits that are passively stable in a narrow region of initial condition parameters, but that are unstable for parameters outside it ‎[5,6]. Therefore, some researchers have proposed stabilizing flight-phase and stance-phase controllers for SLIP running. Flight-phase controllers of the passive SLIP model prepare the swing leg for landing. One bio-inspired strategy uses swing-leg retraction, where the swing leg rotates backwards in the second half of the flight phase ‎[7,‎8]. A dead-beat controller can reject disturbances by adjusting the attack angle or spring stiffness ‎[9]. A control can update each attack angle to be the negative value of the previous take-off angle ‎[10]. Stance-phase controllers require an actuated SLIP architecture in order to add or remove energy from the system. Schmitt et al ‎[11] proposed a stabilizing control law for an active SLIP with a force actuator parallel to the spring. Seipel et al ‎[12] and Ankarali et al ‎[13] proposed control of an active SLIP with torque actuation at the hip and a spring-damper in the leg. Piovan and Byl ‎[14] considered an active SLIP model with a displacement actuator in series to the spring and presented a control strategy. (Note that these works add some control actions to the SLIP model in order to stabilize the natural gait).

The benefits of SLIP include its simplicity and its ability to capture some important dynamics of human running ‎[25]. However, it is totally passive. We propose to use PMB in this work in order to capture the advantages of SLIP while avoiding its passivity disadvantage, although we do compare our results to SLIP. PMB consists of a modified SLIP model where a force actuator replaces the spring (Fig 1). The stance leg in both models endures only compressive forces to keep the foot in contact with the ground. Unlike SLIP, PMB can generate arbitrary running gaits.

For the stance phase trajectories we try a circular arc and polynomials of different degrees. Our method generates running gaits and calculates the necessary leg force profiles in stance phase. The proposed method will work for any desired stance phase path profile consistent with the initial condition.

We defined the stance phase state vector as xs=[r, θ, r˙, θ˙]. Equivalently the velocity can be defined by Vx, Vy, where the vertical velocity can be written as Vy = Vx tan β according to Fig 16, β is the angle of velocity vector V with respect to the horizontal, and θ is the leg angle with respect to the vertical. So instead we can use [r, θ, Vx, β] as the stance phase state vector and [r0, θ0, Vx0, β0] as the stance initial condition. In this section we examine the effects of attack angle θ0 and velocity angle β0 on the efficiency of the running of a biped robot with free leg length of r0 and touch-down speed of Vx0. Thus r0, Vx0 are assumed to be constant with values of 0.95m and 3m/s respectively.

A point mass biped (PMB) model provided the basis for generating arbitrary symmetric trajectories in the stance phase of steady running. To show the generality of the method, a circular-arc trajectory and polynomials of various degrees (all satisfying the initial stance conditions) constituted desired trajectories in stance. The algorithm calculated the required force profile for the leg actuator for each trajectory. We found that paths from polynomials with only two parameters (satisfying initial position and velocity) do not start and end with zero force, which is not desirable. To overcome this problem we added the stance phase initial acceleration to the initial condition, then considered paths with three and four parameters. A degree-4 polynomial with three parameters resulted in a path and force profile very similar to those from a SLIP model, with nearly-equal COT and a maximum leg force 11% less. For a degree-6 polynomial with four parameters, optimizing the maximum leg force resulted in a gait with maximum leg force of 25% less than SLIP. The degree-6 polynomial trajectory can generate biped running gaits with either single-hump or double-hump GRF (previously double-hump profiles was counted as a characteristics of walking ‎[25]). It is worth mentioning that for a defined touch-down velocity, a SLIP model needs a unique attack angle to generate a periodic running gait. However, PMB can generate periodic running gaits with arbitrary touch-down velocities and attack angles (within reasonable limits). We then investigated the effects of attack angle and velocity angle on running efficiency. Choosing the velocity angle as small as possible reduced both COT and maximum leg force. Larger attack angles increased COT and reduced maximum leg force, so there is an optimum value. Using PMB instead of SLIP as a template for multibody biped robots offers some practical advantages: a PMB trajectory can be optimized to overcome real-world limitations such as maximum motor torque, ground coefficient of friction, etc.




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