Two-dimensional team lifting prediction with floating-base box dynamics and grasping force coupling

Abstract

An optimization-based multibody dynamics modeling method is proposed for two-dimensional (2D) team lifting prediction. The box itself is modeled as a floating-base rigid body in Denavit–Hartenberg representation. The interactions between humans and box are modeled as a set of grasping forces which are treated as unknowns (design variables) in the optimization formulation. An inverse-dynamics-based optimization is used to simulate the team lifting motion where the dynamic effort of two humans is minimized subjected to physical and task-based constraints. The design variables are control points of cubic B-splines of joint angle profiles of two humans and the box, and the grasping forces between humans and the box. Analytical sensitivities are derived for all constraints and objective functions including the varying unknown grasping forces. Two numerical examples are successfully simulated: one is to lift a 10 kg box with the center of mass (COM) in the middle, and the other is the same weight box with the COM off the center. The humans’ joint angle, torque, ground reaction force, and grasping force profiles are reported. Reasonable team lifting motion, kinematics, and kinetics are predicted using the proposed multibody dynamic modeling approach and optimization formulation.

Appendix: ZMP and GRF

In this study, an active–passive algorithm [27] is used to calculate ZMP and GRF to obtain the real joint torque for the multibody human system. The algorithm is outlined here as follows:

• (1) Given the state variables $q_{i}, \dot{{q}}_{i}, \ddot{{q}}_{i}$ (design variables) for each DOF, the global resultant active forces ($\mathbf{M}^{o}, \mathbf{F}^{o}$) at the origin in the inertial reference frame (Fig. 1) are obtained from equations of motion without GRF using inverse dynamics.

• (2) After that, the ZMP position is calculated from its definition using the global resultant active force as follows:

  $$ y_{zmp} =0;\quad x_{zmp} =0;\qquad z_{zmp} = \frac{-M_{x}^{o}}{F_{y}^{o}}, $$

  (33)

  where $\mathbf{M}^{o} = [ M_{x}^{o} \ \ 0\ \ 0 ]^{\mathrm{T}}$ and $\mathbf{F}^{o} = [0\ \ F_{y}^{o}\ \ F_{z}^{o} ]^{\mathrm{T}}$. In addition, the two feet are assumed on the level ground.

• (3) After obtaining the ZMP position, the resultant active forces at ZMP ($\mathbf{M}^{zmp}, \mathbf{F}^{zmp}$) are computed using the equilibrium condition as follows:

  $$\begin{aligned} &\mathbf{M}^{zmp} = \mathbf{M}^{o} + \mathbf{F}^{o} \times {}^{o} \mathbf{r}_{zmp},\\ &\mathbf{F}^{zmp} = \mathbf{F}^{o}, \end{aligned}$$

  (34)

  where ${}^{o} \mathbf{r}_{zmp}$ is the ZMP position in the global coordinate system obtained from Eq. (33).

• (4) Then the value and location of GRF are calculated from the equilibrium between the resultant active forces and passive forces at the ZMP:

  $$\begin{aligned} &\mathbf{M}^{GRF} + \mathbf{M}^{zmp} = \mathbf{0}, \\ &\mathbf{F}^{GRF} + \mathbf{F}^{zmp} = \mathbf{0}, \\ &{}^{o} \mathbf{r}_{GRF} - {}^{o} \mathbf{r}_{zmp} = \mathbf{0}, \end{aligned}$$

  (35)

  where $\mathbf{M}^{GRF} = [ M_{x}^{GRF} \ \ 0\ \ 0 ]^{\mathrm{T}}$ and $\mathbf{F}^{GRF} = [0\ \ F_{y}^{GRF}\ \ F_{z}^{GRF} ]^{\mathrm{T}}$.