$$ \dot{x_o^}=\dot{x_o}+D^{-1}[W-W^{}-k(x_o^*-x_o)]...(1) $$

where,

• ẋₒ = Current object velocity (6×1)
• x˙*= Desired object velocity (6×1)
• D = Damping matrix (6×6)
• W = Current wrench on the object (6×1)
• W*= Desired wrench on the object (6×1)
• K = Stiffness matrix (6×6)
• x_o^*= desired object pose
• x_o=current object pose

From the grasp theory, we know that:

$$ J_H \dot{x_o}=G^T\dot{q}\implies \dot{q}=G^{-T}J_H\dot{x_o} $$

• G = Grasp Matrix
• J_H = Hand Jacobian

Take,

$$ G^{-T}J_H =A $$

Now, multiplying A in equation 1, we get:

$$ A\dot{x_o^}=A\dot{x_o}+AD^{-1}[W-W^{}-k(x_o-x_o^{})] \\\implies \dot{q^}=\dot{q}+AD^{-1}[W-W^{}-k(x_o-x_o^{})] $$