Dynamic motion of quadrupedal robots on challenging terrain: a kinodynamic optimization approach

doi:10.1007/s11465-024-0791-5

Front. Mech. Eng.

2024, Vol. 19

Issue (3) : 20 https://doi.org/10.1007/s11465-024-0791-5

Dynamic motion of quadrupedal robots on challenging terrain: a kinodynamic optimization approach

Qi LI^1,², Lei DING^1,², Xin LUO³(

)

¹. Engineering Research Center of Hubei Province for Clothing Information, Wuhan 430200, China
². School of Computer Science and Artificial Intelligence, Wuhan Textile University, Wuhan 430200, China
³. State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China

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Abstract

The dynamic motion of quadrupedal robots on challenging terrain generally requires elaborate spatial–temporal kinodynamic motion planning and accurate control at higher refresh rate in comparison with regular terrain. However, conventional quadrupedal robots usually generate relatively coarse planning and employ motion replanning or reactive strategies to handle terrain irregularities. The resultant complex and computation-intensive controller may lead to nonoptimal motions or the breaking of locomotion rhythm. In this paper, a kinodynamic optimization approach is presented. To generate long-horizon optimal predictions of the kinematic and dynamic behavior of the quadruped robot on challenging terrain, we formulate motion planning as an optimization problem; jointly treat the foot’s locations, contact forces, and torso motions as decision variables; combine smooth motion and minimal energy consumption as the objective function; and explicitly represent feasible foothold region and friction constraints based on terrain information. To track the generated motions accurately and stably, we employ a whole-body controller to compute reference position and velocity commands, which are fed forward to joint controllers of the robot’s legs. We verify the effectiveness of the developed approach through simulation and on a physical quadruped robot testbed. Results show that the quadruped robot can successfully traverse a 30° slope and 43% of nominal leg length high step while maintaining the rhythm of dynamic trot gait.

Keywords quadrupedal robot kinodynamic planning nonlinear optimization challenging terrain whole-body control

Corresponding Author(s): Xin LUO

Issue Date: 01 July 2024

Cite this article:

Qi LI,Lei DING,Xin LUO. Dynamic motion of quadrupedal robots on challenging terrain: a kinodynamic optimization approach[J]. Front. Mech. Eng., 2024, 19(3): 20.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-024-0791-5
https://academic.hep.com.cn/fme/EN/Y2024/V19/I3/20

Fig.1 Trot gait description using an anteroposterior sequence parameterization.

Fig.2 Simplified dynamic model of a quadruped robot. On the basis of the massless leg assumption, the robot is simplified as a torso combined with four massless virtual legs. The virtual leg is formed by connecting the foothold and the center of mass of the torso.

Fig.3 Overview of the locomotion control framework.

Fig.4 Illustration of continuous trajectory generation based on the optimized decision variables.

Fig.5 Definition of swing frame for the swing trajectory generation.

Fig.6 Software framework used in experiments.

Parameter	Value	Unit
μ	0.4
m	13	kg
δ	0.05	m
f_z,max	120	N
$B I$	diag{0.09, 0.157, 0.218}	kg·m²
$B r ¯ 1$	[0.18, −0.13, −0.29]	m
$B r ¯ 2$	[0.18, 0.13, −0.29]	m
$B r ¯ 3$	[−0.18, −0.13, −0.29]	m
$B r ¯ 4$	[−0.18, 0.13, −0.29]	m
c	[0.08, 0.01, 0.1]	m
$K a, i$	100 $1 3 × 3$
$D a, i$	10 $1 3 × 3$
$K q$	3 $1 12 × 12$
$D q$	0.8 $1 12 × 12$
$x l b$	[−inf, −0.1, 0, −0.03, −inf, −0.05, −inf, −1, −1, −1, −inf, −1]^T
$x u b$	[inf, 0.1, 5, 0.03, inf, 0.05, inf, 1, 1, 1, inf, 1]^T
Trot gait parameters	{0.4 s, 0.5, 0.5, 0.5, 0.2 s}

Tab.1 Optimization and control parameters used in simulations and experiments

Parameter	Value
$Q 1$	diag{0.1, 10, 10}
$Q 2$	10 $1 3 × 3$
$Q 3$	1E−4 $1 12 × 12$
$Q 4$	500 $1 12 × 12$

Tab.2 Weighting matrix used in the objective function

Fig.7 Cost of transport of quadruped robot trotting across different slope terrains under different cost functions.

Fig.8 Posture of quadruped robot under different slope terrain simulations: (a) roll angle; (b) pitch angle.

Fig.9 (i–v) Snapshots of the simulation of climbing up a 30° slope terrain.

Fig.10 Simulated CoM and foothold trajectories of the robot trotting on challenging terrain: (a) climbing up a 30° slope; (b) climbing up a 0.13 m high pallet. CoM: center of mass.

Fig.11 Simulation results of climbing up a 30° slope: torso’s (a) pitch angle and (b) forward velocity.

Fig.12 (i–v) Snapshots of the simulation of the robot climbing up a 0.13 m high step.

Fig.13 Simulation results of climbing up a 0.13 m high pallet: torso’s (a) pitch angle and (b) forward velocity.

Fig.14 Comparisons of the KinDyn-Opt and Quad-SDK approaches in terms of trotting up slope and step terrains: (a) trotting up a 20° slope terrain; (b) trotting up a 0.13 m high pallet.

Fig.15 Comparisons of energy consumption and cost of transport (CoT) under two different frameworks: (a) total energy consumption; (b) CoT.

Fig.16 Snapshots of the physical experiment of the robot climbing up (i–v) a 30° slope and (vi–x) a 0.13 m high pallet.

Fig.17 Experimental results of the torso’s orientations: (a) climbing up a 30° slope; (b) climbing up a 0.13 m high pallet.

Fig.18 Experimental results of the RF leg joint angles: (a) climbing up a 30° slope; (b) climbing up a 0.13 m high pallet.

Fig.19 Comparisons of the torso orientation during climbing up a 30° slope: (a) nominal MPC-based method; (b) KinDyn-Opt method. MPC: model predictive controller.

Fig.20 Comparisons of the joint positions, velocities, and torques during climbing up a 30° slope under the nominal MPC-based method and the KinDyn-Opt method: (a) hip joint of the RF leg; (b) knee joint of the RF leg.

Fig.21 Comparisons of the power output of knee joint during climbing up a 30° slope: (a) nominal MPC-based method; (b) KinDyn-Opt method.

Fig.22 Comparisons of the torso orientation during trotting across different height steps: (a) torso roll; (b) torso pitch; (c) torso yaw.

Fig.23 Comparisons of the two different planning methods for climbing up step terrains: (a) forward velocity tracking performance for crossing a 0.13 m high step; (b) CoT of trotting across different height steps. CoT: cost of transport.

Abbreviations
APS	Anteroposterior sequence
CoM	Center of mass
CoT	Cost of transport
KinDyn-Opt	Kinodynamic optimization
MPC	Model predictive controller
NLP	Nonlinear programming problem
RMSE	Root mean square error
Variables
$a i d e s$ , $a ˙ i d e s$ , $a ¨ i d e s$	Desired position, velocity, and acceleration of the ith task
c	Vector of cuboid constraint dimensions
$D, D s t$	Gait cycle duration and stance phase duration, respectively
$F l a g, H l a g, P l a g$	Time lag between the two front feet footfalls, the two hind feet footfalls, and two ipsilateral feet footfalls, respectively
$f i, p i$	Contact force and contact location of ith leg’s foot in inertial frame, respectively
$f (x (t), u (t))$	Equation of motion of the robot
$O i f i$	ith foot contact force expressed in the contact frame
$O i f i x, O i f i y, O i f i z$	x, y, and z components of the ith foot contact force $a i$ , respectively
$f z, max$	Maximum contact force in the z-direction
$g (x (t), u (t))$	Kinematic and dynamic constraints of the robot
$g$	Gravity acceleration vector
$B I, W I$	Inertia tensor of torso expressed in the torso frame and inertial frame, respectively
$J i$	ith task Jacobian
$J i \| p (i)$	Projection of $J i$ into the null space of the previous tasks
$K q$ , $D q$	Joint position and velocity feedback gains, respectively
$K a, i$ , $D a, i$	ith task’s position and velocity feedback gains, respectively
M	Order of the polynomial
$M$ , $C$ , $G$ , $J c, i$	Generalized inertial moment matrix, centripetal and Coriolis force, gravitation force, and ith foot contact Jacobian, respectively
m	Mass of the torso
$N j \| p (j)$	Null space of the $J i \| p (i)$
$n c$	Number of contact points
$O i$	ith foot contact frame
$p c, p ˙ c$	Position and velocity of the torso’s CoM, respectively
$p i x, y, p i z$	x, y, and z positions of ith leg’s foot
$p i, k x, p i, k z$	x and z components of the ith foothold positions at the kth knot point
$S i p ˙ i, k, S i p ˙ i, k + 1$	Velocity of the ith foothold at the initial and final points, respectively
$p (i)$	Previous tasks of task i
$q$	Full configuration of the robot
$Δ q i$	Joint increment related to the ith task iteration
$q i d$ , $q ˙ i d$ , $q ¨ i d$	Joint commands of the ith task
$Q 1$ , $Q 2$ , $Q 3$ , $Q 4$	Positive definite weighting matrices for linear velocities, angular velocities, contact forces, and terminal cost, respectively
$r i, r ˙ i$	Position and velocity of ith leg’s foot relative to the CoM of the torso, respectively
$W R B$	Rotation matrix of the torso frame with respect to the inertial frame
$O i R W$	Rotation matrix, which transforms the contact force vector from the world frame to ith foot contact frame
$S i R W$	Rotation matrix from the inertial frame to the swing frame S₁
$S i$	ith foot swing frame
$S$	Selection matrix
$Δ t$	Discretized time interval
$t s w$	Swing duration of gait
$U (θ)$	Euler-angle rate matrix
$u f$	Input variables in terms of contact forces
W	Positive mechanical power of all active joints
$x, u$	State and input vector of the torso, respectively
$x (T), x t a r g e t$	Final state variables and target state variables, respectively
$x l b, x u b$	Lower and upper bounds of x, respectively
$x 0, x ˙ 0$	Position and velocity of the initial point, respectively
$x T, x ˙ T$	Position and velocity of the final point, respectively
$χ$	Continuous terrain height map
$θ, B ω$	Euler angles and angular velocity of the torso, respectively
$σ i (t)$	Contact state function of ith foot at time t
$μ$	Friction coefficient
$α i$	Angle of rotation of the swing frame S₁ around the y-axis of the inertial frame
$τ f f$	Desired feed-forward joint torques

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