Reduced-order forward dynamics of multiclosed-loop systems

Abstract

In this work, a reduced-order forward dynamics of multiclosed-loop systems is proposed by exploiting the associated inherent kinematic constraints at acceleration level. First, a closed-loop system is divided into an equivalent open architecture consisting of several serial and tree-type subsystems by introducing cuts at appropriate joints. The resulting cut joints are replaced by appropriate constraint forces also referred to as Lagrange multipliers. Next, for each subsystem, the governing equations of motion are derived in terms of the Lagrange multipliers, which are based on the Newton–Euler formulation coupled with the concept of Decoupled Natural Orthogonal Complement (DeNOC) matrices, introduced elsewhere. In the proposed forward dynamics formulation, Lagrange multipliers are calculated sequentially at the subsystem level, and later treated as external forces to the resulting serial or tree-type systems of the original closed-loop system, for the recursive computation of joint accelerations. Note that such subsystem-level treatment allows one to use already existing algorithms for serial and tree-type systems. Hence, one can perform the dynamic analyses relatively quickly without rewriting the complete model of the closed-loop system at hand. The proposed methodology is in contrast to the conventional approaches, where the Lagrange multipliers are calculated together at the system level or simultaneously along with the joint accelerations, both of which incur higher order computational complexities, and thereby a greater number of arithmetic operations. Due to the smaller size of matrices involved in evaluating Lagrange multipliers in the proposed methodology, and the recursive computation of the joint accelerations, the overall numerical performances like computational efficiency, etc., are likely to improve. The proposed reduced-order forward dynamics formulation is illustrated with two multiclosed-loop systems, namely, a 7-bar carpet scrapping mechanism and a 3-RRR parallel manipulator.

Appendices

Appendix A1: Block Jacobian matrices for carpet scrapping mechanism

The analytical expressions for the scalar elements of the block Jacobian matrices obtained in (26) from the loop-closure equations are given below:

$$\mathbf{J}_{1,1} \equiv \left [ \begin{array}{c} a_1s_1 \\ a_1c_1 \\ \end{array} \right ], \quad \mathbf{J}_{1,2} \equiv \left [ \begin{array}{c@{\quad}c} -a_2s_2-a_3s_3 & -a_3s_3\\ -a_2c_2-a_3c_3 & -a_3c_3\\ \end{array} \right ], \quad \mathbf{J}_{2,2} \equiv \left [ \begin{array}{c@{\quad}c} -a_2s_2+a_{39}s_3 & a_{39}s_3\\ -a_2c_2-a_{39}c_3 & -a_{39}c_3\\ \end{array} \right ], $$

$$\mathbf{J}_{2,3} \equiv \left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} a_{59}s_5+(a_{46}-a_{56})s_4 & a_{59}s_5 & 0 & 0\\ -a_{59}c_5-(a_{46}-a_{56})c_4 & -a_{59}c_5 & 0 & 0\\ \end{array} \right ], $$

$$\mathbf{J}_{3,3} \equiv \left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} -a_{46}s_4-a_{67}s_6 & & &\\ -a_{97}s_7+a_{59}s_5+(a_{46}-{a}_{56})s_4 & a_{59}s_5 & -a_{67}s_6-a_{97}s_7 & -{a}_{97}s_7\\ a_{46}c_4+a_{67}c_6 & & \\ +a_{97}c_7-a_{59}c_5-(a_{46}-{a}_{56})c_4 & -a_{59}c_5 & a_{67}c_6+a_{97}c_7 & a_{97}c_7\\ \end{array} \right ] $$

where a _i or a _ij represents link length (kinematic parameters), as shown in Fig. 9(a), and s _i and c _i represent sinα _i and cosα _i , respectively. The relations between α _i ’s and θ _i ’s are given by α ₁=θ ₁, α ₂=θ ₂, α ₃=π+θ ₂+θ ₃, α ₄=θ ₄, α ₅=θ ₅+θ ₄−2π, α ₆=θ ₆+θ ₄−2π and α ₇=θ ₇+θ ₆+θ ₄−4π, where α _i represents the angle subtended by a link with the horizontal line and θ _i ’s are the relative joint angles between the links at the joint locations indicated in the subscripts of θ _i , as shown in Fig. 4.

Fig. 9

Kinematic architectures

Appendix A2: Block Jacobian matrices for 3-RRR parallel manipulator

The analytical expressions for the scalar elements of the block Jacobian matrices obtained in (37) from the loop-closure equations are given below:

$$\mathbf{J}_{1,1} \equiv \left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} -a_1s_1-b_1s_4-d_1s_7+b_2s_5 & -b_1s_4-d_1s_7+b_2s_5 & -d_1s_7+b_2s_5 & b_2s_5 & 0\\ a_1c_1+b_1c_4-d_1c_7-b_2c_5 & \quad b_1c_4-d_1c_7-b_2c_5 & d_1c_7-b_2c_5 & -b_2c_5 & 0\\ \end{array} \right ], $$

$$\mathbf{J}_{2,1} \equiv \left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} -a_1s_1-b_1s_4-d_1s_7^\prime+b_3s_6 & -b_1s_4-d_1s_7^\prime+b_3s_6 & -d_1s_7^\prime+b_3s_6 & 0 & b_3s_6\\ a_1c_1+b_1c_4+d_1c_7^\prime-b_3c_6 & b_1c_4+d_1c_7^\prime-b_3c_6 & d_1c_7^\prime-b_3c_6 & 0 & -b_3c_6\\ \end{array} \right ], $$

$$\mathbf{J}_{1,2} \equiv \left [ \begin{array}{c} a_2s_2\\ -a_2c_2\\ \end{array} \right ], \quad \mathbf{J}_{1,3} \equiv \mathbf{J}_{2,2} \equiv \left [ \begin{array}{c} 0\\ 0\\ \end{array} \right ], \quad \mathbf{J}_{2,3} \equiv \left [ \begin{array}{c} a_3s_3\\ -a_3c_3\\ \end{array} \right ] $$

where a _i , b _i , and d _i represent the kinematic parameters of the 3-RRR parallel manipulator shown in Fig. 9(b). The relations between α _i ’s and θ _i ’s are given by α ₁=θ ₁, α ₂=θ ₆, α ₃=θ ₇, α ₄=θ ₁+θ ₂, α ₅=θ ₁+θ ₂+θ ₃+θ ₄−π, α ₆=θ ₁+θ ₂+θ ₃+θ ₅−(π−π/3), α ₇=θ ₁+θ ₂+θ ₃ and $\alpha_{7}^{\prime}= \alpha_{7} + \pi{/3}$.

Appendix A3: Block Gaussian elimination of (10)

After carrying out the forward block Gaussian elimination of (10), we arrive at

$$ \left [ \begin{array}{c@{\quad}c} \mathbf{I} & \mathbf{J}^T \\ \mathbf{O} & -\mathbf{JI}^{-1}\mathbf{J}^T\\ \end{array} \right ] \left [ \begin{array}{c} \ddot{\mathbf{q}}\\ -\boldsymbol {\lambda }\\ \end{array} \right ] = \left [ \begin{array}{c} \boldsymbol {\upvarphi }\\ -\dot{\mathbf{J}}\dot{\mathbf{q}} - \mathbf{JI}^{-1}\boldsymbol {\upvarphi }\\ \end{array} \right ] $$

(47)

The vector of Lagrange multipliers is thus obtained as

$$ \boldsymbol {\lambda }= -\bigl(\mathbf{JI}^{-1}\mathbf{J}^T \bigr)^{-1}\bigl(\dot{\mathbf{J}}\dot {\mathbf{q}} + \mathbf{JI}^{-1}\boldsymbol {\upvarphi }\bigr) $$

(48)

which is same as (12).

Appendix A4: Block Gaussian elimination of (23) for the 3-RRR parallel manipulator

Using the expressions of (42), (23) for the 3-RRR parallel manipulator is reproduced here as

$$ \left [ \begin{array}{c@{\quad}c} \overline{\mathbf{I}}_{1,1} & \overline{\mathbf{I}}_{2,1}^T \\ \overline{\mathbf{I}}_{2,1} & \overline{\mathbf{I}}_{2,2} \\ \end{array} \right ] \left [ \begin{array}{c} \boldsymbol {\lambda }_1\\ \boldsymbol {\lambda }_2\\ \end{array} \right ] = \left [ \begin{array}{c} \overline{\boldsymbol {\Psi }}_1\\ \overline{\boldsymbol {\Psi }}_2\\ \end{array} \right ] $$

(49)

Applying the forward block Gaussian elimination using the block pivot as $\overline{\mathbf{I}}_{2,1}\overline{\mathbf{I}}_{1,1}^{-1}$, on (49) the following block upper triangular matrix on the LHS is obtained

$$ \left [ \begin{array}{c@{\quad}c} \overline{\mathbf{I}}_{1,1} & \overline{\mathbf{I}}_{2,1}^T \\ \mathbf{O}&\overline{\mathbf{I}}_{2,2}-\overline{\mathbf{I}}_{2,1}\overline{\mathbf{I}}_{1,1}^{-1}\overline{\mathbf{I}}_{2,1}^T \\ \end{array} \right ] \left [ \begin{array}{c} \boldsymbol {\lambda }_1\\ \boldsymbol {\lambda }_2\\ \end{array} \right ] = \left [ \begin{array}{c} \overline{\boldsymbol {\Psi }}_1\\ \overline{\boldsymbol {\Psi }}_2 - \overline{\mathbf{I}}_{2,1}\overline{\mathbf{I}}_{1,1}^{-1}\overline{\boldsymbol {\Psi }}_1\\ \end{array} \right ] $$

(50)

Using the backward substitutions, one can then solve easily for λ ₂ and λ ₁, which are given by

$$\begin{aligned} \boldsymbol {\lambda }_2 \equiv&\tilde{\mathbf{I}}_{2,2}^{-1} \widetilde{\boldsymbol {\Psi }}_2 \end{aligned}$$

(51)

$$\begin{aligned} \boldsymbol {\lambda }_1 \equiv&\tilde{\mathbf{I}}_{1,1}^{-1} \bigl(\overline{\boldsymbol {\Psi }}_1 - \overline{\mathbf{I}}_{2,1}^T \boldsymbol {\lambda }_2\bigr) \end{aligned}$$

(52)

where $\tilde{\mathbf{I}}_{2,2} \equiv\overline{\mathbf{I}}_{2,2} - \overline{\mathbf{I}}_{2,1}\overline{\mathbf{I}}_{1,1}^{-1}\overline{\mathbf{I}}_{2,1}^{T}$ and $\widetilde{\boldsymbol {\Psi }}_{2} \equiv \overline{\boldsymbol {\Psi }}_{2} - \overline{\mathbf{I}}_{2,1}\overline{\mathbf{I}}_{1,1}^{-1}\overline{\boldsymbol {\Psi }}_{1}$.

Appendix A5: Block Gaussian elimination of the subsystem-level representation of (10) for the 3-RRR parallel manipulator

To illustrate that the results of (51)–(52) can also be obtained from the fundamental formulation given by (10), by using the subsystem-level expressions for the 3-RRR parallel manipulator, i.e., (37)–(40), (10) is rewritten as

$$ \left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \mathbf{I}_1 & \mathbf{O} & \mathbf{O} & \mathbf{J}_{1,1}^T & \mathbf{J}_{2,1}^T\\ \mathbf{O} & \mathbf{I}_2 & \mathbf{O} & \mathbf{J}_{1,2}^T & \mathbf{O}\\ \mathbf{O} & \mathbf{O} & \mathbf{I}_3 & \mathbf{O} & \mathbf{J}_{2,3}^T\\ \mathbf{J}_{1,1} & \mathbf{J}_{1,2} & \mathbf{O} & \mathbf{O} & \mathbf{O}\\ \mathbf{J}_{2,1} & \mathbf{O} & \mathbf{J}_{2,3} & \mathbf{O} & \mathbf{O}\\ \end{array} \right ] \left [ \begin{array}{c} \ddot{\mathbf{q}}_1\\ \ddot{\mathbf{q}}_2\\ \ddot{\mathbf{q}}_3\\ -\boldsymbol {\lambda }_1\\ -\boldsymbol {\lambda }_2\\ \end{array} \right ] = \left [ \begin{array}{c} \boldsymbol {\upvarphi }_1\\ \boldsymbol {\upvarphi }_2\\ \boldsymbol {\upvarphi }_3\\ \boldsymbol {\Psi }_1\\ \boldsymbol {\Psi }_2\\ \end{array} \right ] $$

(53)

Applying the block Gaussian elimination, the final upper block triangular on the LHS is given by

$$ \left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \mathbf{I}_1 & \mathbf{O} & \mathbf{O} & \mathbf{J}_{1,1}^T & \mathbf{J}_{2,1}^T\\ \mathbf{O} & \mathbf{I}_2 & \mathbf{O} & \mathbf{J}_{1,2}^T & \mathbf{O}\\ \mathbf{O} & \mathbf{O} & \mathbf{I}_3 & \mathbf{O} & \mathbf{J}_{2,3}^T\\ \mathbf{O} & \mathbf{O} & \mathbf{O} & -\overline{\mathbf{I}}_1^{11}-\overline{\mathbf{I}}_2^{11} & -\overline{\mathbf{I}}_1^{12}\\ \mathbf{O} & \mathbf{O} & \mathbf{O} & \mathbf{O} & -\overline{\mathbf{I}}_1^{22}-\overline{\mathbf{I}}_3^{22}+\overline{\mathbf{I}}_1^{21}(\overline{\mathbf{I}}_1^{11} + \overline{\mathbf{I}}_2^{11})^{-1}(\overline{\mathbf{I}}_1^{21})^T\\ \end{array} \right ] \left [ \begin{array}{c} \ddot{\mathbf{q}}_1\\ \ddot{\mathbf{q}}_2\\ \ddot{\mathbf{q}}_3\\ -\boldsymbol {\lambda }_1\\ -\boldsymbol {\lambda }_2\\ \end{array} \right ] = \left [ \begin{array}{c} \boldsymbol {\upvarphi }_1\\ \boldsymbol {\upvarphi }_2\\ \boldsymbol {\upvarphi }_3\\ \widetilde{\boldsymbol {\Psi }}_1\\ \widetilde{\boldsymbol {\Psi }}_2\\ \end{array} \right ] $$

(54)

where $\widetilde{\boldsymbol {\Psi }}_{1}\equiv \boldsymbol {\Psi }_{1} - \mathbf{J}_{1,1}\mathbf{I}_{1}^{-1}\boldsymbol {\upvarphi }_{1} - \mathbf{J}_{1,2}\mathbf{I}_{2}^{-1} \boldsymbol {\upvarphi }_{2}$ and $\widetilde{\boldsymbol {\Psi }}_{2}\equiv \boldsymbol{\Psi}_{2} - \mathbf{J}_{2,1}\mathbf{I}_{1}^{-1} \boldsymbol {\upvarphi }_{1} - \mathbf{J}_{2,3}\mathbf{I}_{3}^{-1}\boldsymbol {\upvarphi }_{3} - \mathbf{I}_{1}^{21} (\mathbf{I}_{1}^{11} + \mathbf{I}_{2}^{11})^{-1}\widetilde{\boldsymbol {\Psi }}_{1}$. The Lagrange multipliers are computed next using backward substitutions as

$$ \boldsymbol {\lambda }_2 \equiv\bigl[\overline{\mathbf{I}}_1^{22} + \overline{\mathbf{I}}_3^{22} - \overline{ \mathbf{I}}_1^{12}\bigl(\overline{\mathbf{I}}_1^{11} + \overline{\mathbf{I}}_2^{11}\bigr)^{-1}\bigl( \overline{\mathbf{I}}_1^{21}\bigr)^T \bigr]^{-1}\widetilde{\boldsymbol {\Psi }}_2 $$

(55)

$$ \boldsymbol {\lambda }_1 \equiv\bigl(\overline{\mathbf{I}}_1^{11} + \overline{\mathbf{I}}_2^{11}\bigr)^{-1}\bigl[ \overline{\boldsymbol {\Psi }}_1 - \bigl(\overline{\mathbf{I}}_1^{21} \bigr)^T\boldsymbol {\lambda }_2\bigr] $$

(56)

Substituting the expressions of (43) in (55)–(56), we get exactly the same expressions for λ ₁ and λ ₂ to that obtained in (51)–(52) and reported in (45)–(46). Note here that one could continue with the back substitutions to solve for the joint accelerations required in the forward dynamics. This is, however, not recommended here, mainly due to the fact that one is not able to exploit the already existing algorithms for the tree-type systems, which may be of recursive in nature, e.g., ReDySim of [30], with good efficiency and numerical stability.