Dynamics of serial kinematic chains with large number of degrees-of-freedom

Abstract

This paper investigates the dynamic behaviour of serial chains with degrees-of-freedom (DOF) as large as 100,000. A recursive solver called Recursive Dynamic Simulator (ReDySim), based on the Newton–Euler formulation and the Decoupled Natural Orthogonal Complement (DeNOC) matrices, was used to simulate the dynamics of these systems. Planar, as well as spatial motions of chains with moderate- (DOF≤1,000), large- (1,000<DOF≤10,000), and huge- (DOF>10,000) DOF were simulated. The results were validated by several means, such as comparisons with reported results wherever available, results obtained from commercial software, energy checks, etc. The study shows that ReDySim is capable of analysing serial chains of huge-DOF with acceptable numerical accuracy. The scheme is found to be numerically stable as well as computationally efficient, owing to the linear-time computation of the joint accelerations. Numerical studies were conducted to establish the theoretical basis for better performance of the proposed ReDySim solver. With the demonstrated capabilities of ReDySim, it may be found suitable for a large number of applications involving serial systems with huge-DOF.

Appendix

1.1 A.1 Recursive O(n) forward dynamics

In this section, outline of the recursive O(n) forward dynamics algorithm implemented in Recursive Dynamics Simulator (ReDySim) solver and used for the simulation of chains with large-DOF are presented. The algorithm is based on the UDU ^T decomposition of the GIM [27] using analytical Reverse Gaussian Elimination, denoted by RGE-A. The forward dynamics essentially involves the calculation of joint accelerations from Eq. (9) as

$$\begin{aligned} \ddot{\mathbf{q}} = \mathbf{I}^{ - 1}\boldsymbol{\varphi},\quad\mbox{where } \boldsymbol{\varphi}= \boldsymbol{\tau}- \mathbf{C}\dot{\mathbf{q}}. \end{aligned}$$

(A.1)

Using the decomposition I=UDU ^T, the equations of motion, Eq. (A.1), are rewritten as follows:

$$\begin{aligned} \mathbf{UDU}^{T}\ddot{\mathbf{q}} = \boldsymbol{\varphi}, \end{aligned}$$

(A.2)

where U and D, the n×n block upper-triangular and diagonal matrices, respectively. They have the following representations:

$$\begin{aligned} \mathbf{U} \equiv \left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \mathbf{1} & \mathbf{U}_{1,2} & \ldots & \mathbf{U}_{1,n_{l}} \\ & \mathbf{1} & & \vdots \\ & & \ddots & \mathbf{U}_{n_{l} -1,n_{l}} \\ \mathbf{O}\mbox{'s} & & & \mathbf{1} \end{array} \right ]\quad\mbox{and} \quad \mathbf{D}\equiv \left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \hat{\mathbf{I}}_{1} & & \mathbf{O}\mbox{'s} \\ &\ddots & \\ \mathbf{O}\mbox{'s} & & \hat{\mathbf{I}}_{n_{l}} \end{array} \right ]. \end{aligned}$$

(A.3)

In Eq. (A.3), U _k,j and $\hat{\mathbf{I}}_{k}$ are the z _k ×z _j and z _k ×z _k matrices, which are scalars for one-DOF jointed chains, as shown in [26, 27]. The matrices U _k,j and $\hat{\mathbf{I}}_{k}$ are given by

$$ \begin{aligned} \mathbf{U}_{k,j} &\equiv \mathbf{P}_{k}^{T}\mathbf{A}_{j,k}^{T}\boldsymbol{\Psi}_{j},\\ \hat{\mathbf{I}}_{k} &\equiv \mathbf{P}_{k}^{T}\hat{\boldsymbol{\Psi}}_{k}, \end{aligned} $$

(A.4)

where the 6×z _k matrices Ψ _k and $\hat{\boldsymbol{\Psi}}_{k}$ are as follows:

$$\begin{aligned} \boldsymbol{\Psi}_{k} = \hat{\boldsymbol{\Psi}}_{k}\hat{ \mathbf{I}}_{k}^{ - 1} \quad\mbox{and}\quad \hat{\boldsymbol{\Psi}}_{k} = \hat{ \mathbf{M}}_{k}\mathbf{P}_{k}. \end{aligned}$$

(A.5)

In Eq. (A.5), the 6×6 matrix $\hat{\mathbf{M}}_{k}$ contains the mass and inertia properties of the articulated-body #k [23]. The matrix $\hat{\mathbf{M}}_{k}$ is computed as follows:

$$\begin{aligned} \hat{\mathbf{M}}_{k} = \mathbf{M}_{k} + \mathbf{A}_{k + 1,k}^{T}\hat{\mathbf{M}}_{k + 1,k + 1} \mathbf{A}_{k + 1,k}, \end{aligned}$$

(A.6)

where $\hat{\mathbf{M}}_{n} \equiv \mathbf{M}_{n}$. Moreover, the 6×6 matrix $\hat{\mathbf{M}}_{k + 1,k + 1}$ in Eq. (A.6) is obtained as

$$\begin{aligned} \hat{\mathbf{M}}_{k + 1,k + 1} = \hat{\mathbf{M}}_{k + 1} - \hat{\boldsymbol{\Psi}}_{k + 1}\boldsymbol{\Psi}_{k + 1}^{T}. \end{aligned}$$

(A.7)

The joint accelerations are then obtained using three sets of linear algebraic equations, namely

$$\begin{aligned} \mathrm{(i)}\, \ \ \quad &\mathbf{U}\hat{\boldsymbol{\varphi}} = \boldsymbol{\varphi},\quad\mbox{where } \hat{\boldsymbol{\varphi}} \equiv \mathbf{DU}^{T}\ddot{\mathbf{q}}, \\ \mathrm{(ii)}\,\ \quad &\mathbf{D}\tilde{\boldsymbol{\varphi}} = \hat{\boldsymbol{\varphi}},\quad\mbox{where } \tilde{\boldsymbol{\varphi}} \equiv \mathbf{U}^{T}\ddot{\mathbf{q}}, \\ \mathrm{(iii)}\quad& \mathbf{U}^{T}\ddot{\mathbf{q}} = \tilde{\boldsymbol{\varphi}}. \end{aligned}$$

(A.8)

From Eq. (A.8), the recursive algorithm, comprising of the backward recursions arising from steps (i) and (ii), and forward recursions, arising from step (iii), is summarised in Table 12. It is assumed in Table 12 that the vector φ, comprised of the external joint torques and those due to Coriolis and centripetal accelerations, is calculated using an O(n) inverse dynamic algorithm after setting $\ddot{\boldsymbol{\theta}} = \mathbf{0}$, as shown in [29]. The effect of gravity can be taken into consideration during the inverse dynamics calculations by adding the gravitational acceleration vector into the twist-rate of the first link. Table 12 explains the steps involved in forward dynamic calculations.

Table 12 Forward dynamic algorithm [23]

1.2 A.2 Geometric interpretations of GE and RGE

After GE-N, which is denoted here with simply GE as no numerical aspect is emphasised, the GIM of Eq. (10a) is converted to an equivalent upper triangular matrix during the annihilations or sweep-out phase, whereas the solution of the joint accelerations is obtained using backward substitution as

$$\begin{aligned} \mathrm{GE}: \quad\mathbf{I}\ddot{\mathbf{q}} = \boldsymbol{\varphi}\quad \mathop{\Rightarrow}^{\mathrm{Phase~1}}\quad\mathbf{U}\ddot{\mathbf{q}} = \bar{\boldsymbol{\varphi}}\quad \mathop{\Rightarrow}^{\mathrm{Phase~2}}\quad\ddot{\mathbf{q}} = \mathbf{U}^{ - 1}\bar{\boldsymbol{\varphi}}. \end{aligned}$$

(A.9)

The matrix representing multipliers in the sweep-out phase is a lower triangular matrix such that I=LU [25]. Therefore, after GE of the GIM, κ ₂(L) and κ ₂(U) will give an estimate of the accuracy of the sweep-out and backward substitution phases, respectively. The matrices L and U, corresponding to the GIM I(0) in Table 9, and their condition numbers are shown in Table 13. On the contrary, in RGE-N or RGE-A, which are denoted here by RGE only because no numerical or analytical aspect is emphasised, the GIM of Eq. (10a) is converted into an equivalent lower triangular matrix during the sweep-out phase, and the solution of the joint accelerations are obtained using forward substitution as

$$\begin{aligned} \mathrm{RGE}:\quad \mathbf{I}\ddot{\mathbf{q}} = \boldsymbol{\varphi}\quad \mathop{\Rightarrow}^{\mathrm{Phase~1}}\quad \mathbf{L}'\ddot{\mathbf{q}} = \boldsymbol{\varphi}'\quad \mathop{\Rightarrow}^{\mathrm{Phase~2}}\quad\ddot{\mathbf{q}} = \mathbf{L}'^{ - 1}\boldsymbol{\varphi}'. \end{aligned}$$

(A.10)

Table 13 Condition number of the matrices resulting out GE and RGE

In RGE, the matrix representing multipliers of the sweep-out phase is an upper triangular matrix such that I=U′L′. The matrices U′ and L′ and their condition numbers are also given in Table 13.

It can be seen from Table 13 that the condition number of L after GE is κ ₂(L)=2.17, which ensures the accuracy of the sweep-out phase. However, the condition number of U, κ ₂(U)=223.11, makes backward substitution phase inaccurate or sensitive to small error in the right-hand side, as shown in Table 9. On the other hand, in RGE, κ ₂(U′)=21.46 while κ ₂(L′)=15.15. Hence, both the phases have low sensitivity to deviations in the right-hand sides, thereby, giving better solution to the system of linear equations.

This above fact was also supported in [36], where it was shown geometrically that the use of partial pivoting to achieve stability in sweep-out phase may reorient the planes such that it makes substitution phase highly inaccurate. Hence, geometric analysis of both GE and RGE was performed. Figure 20(a–c) show the planes of the original system, $\mathbf{I}\ddot{\mathbf{q}} = \boldsymbol{\varphi}$, and reoriented systems $\mathbf{U}\ddot{\mathbf{q}} = \bar{\boldsymbol{\varphi}}$ due to GE of the GIM and $\mathbf{L}'\ddot{\mathbf{q}} = \boldsymbol{\varphi}\mathbf{ }\boldsymbol{'}$ due to RGE. Figure 20(a) shows that the original system is poorly oriented as the angles between them are small. The GE does not make the situation better as planes 2 and 3 are still poorly oriented, as shown in Fig. 20(b). As GE starts backward substitution with calculation of $\ddot{q}_{3}$, small deviation in $\ddot{q}_{3}$ (orientation of plane 3) causes large changes in $\ddot{q}_{2}$ and $\ddot{q}_{1}$. On the other hand, RGE of the GIM, as shown in Fig. 20(c), makes the orientation of the plane favourable for forward substitution as the angle between the planes is higher. Hence, small deviations in $\ddot{q}_{1}$ do not result into large changes in $\ddot{q}_{2}$ and $\ddot{q}_{3}$.