Sagittal SLIP-anchored task space control for a monopode robot traversing irregular terrain

doi:10.1007/s11465-019-0569-3

Front. Mech. Eng.

2020, Vol. 15

Issue (2) : 193-208 https://doi.org/10.1007/s11465-019-0569-3

RESEARCH ARTICLE

Sagittal SLIP-anchored task space control for a monopode robot traversing irregular terrain

Haitao YU(

), Haibo GAO, Liang DING, Zongquan DENG

State Key Laboratory of Robotics and Systems, Harbin Institute of Technology, Harbin 150001, China

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Abstract

As a well-explored template that captures the essential dynamical behaviors of legged locomotion on sagittal plane, the spring-loaded inverted pendulum (SLIP) model has been extensively employed in both biomechanical study and robotics research. Aiming at fully leveraging the merits of the SLIP model to generate the adaptive trajectories of the center of mass (CoM) with maneuverability, this study presents a novel two-layered sagittal SLIP-anchored (SSA) task space control for a monopode robot to deal with terrain irregularity. This work begins with an analytical investigation of sagittal SLIP dynamics by deriving an approximate solution with satisfactory apex prediction accuracy, and a two-layered SSA task space controller is subsequently developed for the monopode robot. The higher layer employs an analytical approximate representation of the sagittal SLIP model to form a deadbeat controller, which generates an adaptive reference trajectory for the CoM. The lower layer enforces the monopode robot to reproduce a generated CoM movement by using a task space controller to transfer the reference CoM commands into joint torques of the multi-degree of freedom monopode robot. Consequently, an adaptive hopping behavior is exhibited by the robot when traversing irregular terrain. Simulation results have demonstrated the effectiveness of the proposed method.

Keywords legged robots spring-loaded inverted pendulum task space control apex return map deadbeat control irregular terrain negotiation

Corresponding Author(s): Haitao YU

Just Accepted Date: 11 February 2020 Online First Date: 09 March 2020 Issue Date: 25 May 2020

Cite this article:

Haitao YU,Haibo GAO,Liang DING, et al. Sagittal SLIP-anchored task space control for a monopode robot traversing irregular terrain[J]. Front. Mech. Eng., 2020, 15(2): 193-208.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-019-0569-3
https://academic.hep.com.cn/fme/EN/Y2020/V15/I2/193

Fig.1 Schematic of the sagittal SLIP-anchored task space control architecture. SLIP: Spring-loaded inverted pendulum.

Fig.2 Illustration of the sagittal SLIP model. (a) Coordinates and variable definitions; (b) entire gait cycle, including flight and stance sub-phase.

Fig.3 Composition of ARM, including the three sub-maps of P_fd, P_st, and P_fu.

Fig.4 Schematic of the variable stiffness policy with piecewise constant profile for the compression and decompression phases. (a) Division of the stance phase, with the instant stiffness variation at the bottom defined as the maximum leg compression; (b) variable stiffness spring with piecewise-constant leg stiffness.

Algorithm: Determination of the touchdown angle and the leg stiffness in solving problem Eq. (7)

Input:

The initial apex state S₀

The target apex state S_d

The initial leg stiffness k_s

Output:

The touchdown angle α_TD

The leg stiffness k_c and k_d

1. Initialize the current leg stiffness

k c ← k s

2. Compute the energy variation

Δ E

by using Eq. (9)

3. for α_TD = π/4 to π/2 do

4. Compute the approximation of the leg length at BM

r ˜ BM

k d ← k c + 2 Δ E / (r − r ˜ BM) 2

6. Compute sub-maps

P fd

P ˜ st

P fu

7. Compute the A²RM

P ˜ ← P fd ∘ P ˜ st ∘ P fu

8. Compute the predicted apex state

S ˜ n + 1 ← P ˜ (S 0, (α TD, k c, k d) T)

α TD = arg min ‖ S d − P ˜ (S 0, (α TD, k c, k d) T) ‖

10. end for

11. redo Steps 3 and 4

12. return α_TD, k_c, and k_d

13 Update the leg stiffness for the coming compression sub-phase with

k s ← k d

14. end algorithm

Tab.1

Fig.5 Hopping with constant absolute altitude when traversing irregular terrains in the SLIP model. Colored curves represent CoM trajectories of different terrain irregularities. y₀ and y_d represents the initial and target hopping height, respectively.

Fig.6 Rigid model of the monopode robot. (a) Leg configuration and relevant parameters; (b) entire hopping gait cycle with sub-phase division. AoT α_TD and virtual equivalent leg r_eq are the corresponding parameters used in the SLIP model.

Fig.7 FSM for monopode robot hopping.

Tab.2 Model parameters of the monopode robot in the simulation

Tab.3 Model parameters of the sagittal SLIP model in the simulation

Fig.8 Snapshots of the CoM trajectory of the monopode robot to represent periodic hopping on a flat surface.

Fig.9 Simulation results of selected variables for the monopode robot on constant apex height tracking. (a) Evolution of joint angles q₁ and q₂; (b) apex height y_a and AoT α_TD from stride to stride.

Fig.10 Snapshots of the CoM trajectory of the monopode robot executing target apex tracking on a flat surface.

Fig.11 Simulation results of selected variables of the monopode robot on variable apex state tracking. (a) Evolution of joint angles q₁ and q₂; (b) apex height and velocity from stride to stride.

Fig.12 Snapshots of the CoM trajectory of the monopode robot traversing an irregular terrain.

Fig.13 Simulated horizontal velocity and vertical height of CoM of the monopode robot.

Fig.14 Phase portrait of joint angles q₁ and q₂ of the robot traversing an irregular terrain. Arrows represent the evolution of the selected variables over time.

Tab.4 General feature comparison between the traditional SLIP controller and the proposed controller

Fig.15 Comparative simulation results of the monopode robot equipped with the traditional and proposed controllers. Shaded areas represent convergence duration for each case.

Fig.16 ARM of the SLIP model equipped with traditional fixed AoT policy. Shaded areas represent the unfeasible region in which stumbling occurs with insufficient initial releasing height y_a(i)<r₀sinα_TD. Stable and unstable fixed points are colored green and red, respectively. The partial view shows the convergence process after initial release at AoT α_TD = 64°.

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