The MSHAKE algorithm, which is applied by modifying the leapfrog algorithm to include forces of constraint, computes an initial estimate of constraint forces, then iteratively corrects the constraint forces required to maintain the fixed distances.
The two subsystems are primarily coupled through the four forces of constraint Γ.
The variable v 1 represents the moment of the forces of constraint v_{1} = R_{p} varGamma.
The large gain is K and v 1 describes the dynamic effect of the forces of constraint on the platform support forces acting on the system.
Let us assume, for simplicity and ease of notation, that the only forces of constraint remaining in the system are Γ id.
A major step in the above development is that the computation of forces of constraint has been extended to systems described by a combined system of kinematics and dynamics.
The system could also have been described by combining constraints and forces of constraint begin{aligned} & CC = left[ {C_{1}^{prime },; - C_{2}^{prime } } right]^ & varGamma = varGamma_{1} end{aligned} (4) Fig. 1 The planar one-segment biped and a particular planar platform subsystem.
Further, it is assumed that the constraints C 1 can be separated into two parts as shown in Eq. (5): C_{1} = C varTheta ) + D varPhi ) (5 For the computation of the forces of constraint Γ1, it is necessary to differentiate equation (5) twice with respect to time.
The main difference between the all dynamic and the combined kinematic dynamic systems is that, for the latter system, the forces of constraint are functions of the kinematic accelerations, velocity and positions in addition to being functions of the state and the remaining inputs.
In practical terms, this means that the controlled subsystem must, through its sensory apparatus, measure the forces of constraint Γ and, from these measurements and possibly its actuators dynamics that may be sensitive to the platform motion, extract the controlling subsystem's positions, velocities and acceleration variables.
With these clarifications, the equations for the combined system are: I_{1} (Z_{1} )ddot{Z}_{1} + B(Z_{1},dot{Z}_{1} )dot{Z}_{1} - gG(Z_{1} ) = W_{1} U_{i} + VvarGamma_{id} + V_{1} (beta,dot{beta },ddot{beta }) (11)From Eq. (11), one can derive the required forces of constraint Γ id similar to Eq. (8).
