Dynamics analysis, offline–online tuning and identification of base inertia parameters for the 3-DOF Delta parallel robot under insufficient excitations

Abstract

In this paper, the dynamics model of the 3-Degrees-Of-Freedom (DOF) Delta parallel manipulator is elaborated by means of a Screw-based formulation of the virtual works method in a linear regression form. Moreover, a reduced dynamics model is obtained by exploiting the singular value decomposition, leading to the determination of the base inertial parameters. This method is an offline tuning approach for which an optimized path containing several harmonics is required to establish the desired bandwidth. Additionally, this paper presents an algorithm to address the singularity problem by mapping the regressors to an orthogonal space and obtaining an optimal set of base parameters in an online manner, namely the online base inertial parameter tuning method. Thereafter, through a set of simulations, the effectiveness of the proposed method has been illustrated for different paths. Finally, a change in the value of an inertial parameter of the system, which can be regarded as a disturbance, is exerted and the obtained results reveal that the proposed method can compensate for extrinsic mass effect. In summary, it can be deduced that the the proposed approach is also able to identify the changed model under insufficient excitations for real-time purposes.

Appendix

1.1 Wrenches of Inertia with respect to the link coordinate frames

Inertial wrenches of a body, defined in Eq. (36), have been determined in Eq. (37) w.r.t. the center of gravity of the corresponding link. However, from the theory of second order infinitesimal rotations, one can write acceleration of the center of mass of each link relative to another point on the corresponding link. Furthermore, from Euler’s rotation equation of motion, it can be written relative to an arbitrary point on the rigid body. Therefore, taking these concepts into account, Eq. (37) would be:

$$\begin{aligned} {\mathscr {F}}_{G_{i}}^{\mathrm {inertia}} = \left[ \begin{array}{c}{m \left( \dot{\mathbf {v}}_{i}+\pmb {\omega }_{i} \times \left( \pmb {\omega }_{i} \times \mathbf {r}_{i}\right) +\dot{\pmb {\omega }}_{i} \times \mathbf {r}_{i}\right) } \\ {\mathbf {I}_{\mathrm {G}_{i}} \dot{\pmb {\omega }}_{i}+\pmb {\omega }_{i} \times \mathbf {I}_{\mathrm {G}_{i}} \pmb {\omega }_{i}+m \left( \mathbf {r}_{i} \times \dot{\mathbf {v}}_{\mathrm {G}_{i}}\right) }\end{array}\right] . \end{aligned}$$

(88)

As it is can be observed from the above, the two equations constitute the wrenches of inertia; first one as Newton’s second law of motion and the second one as Euler’s rotation equation; both defined w.r.t. an arbitrary location, namely, center of the links’ coordinate frames. However, the second equation contains some terms regarding the center of mass which should be eliminated. Thus, the inertial tensor of each link has been written w.r.t. the center of its own link coordinate frame:

$$\begin{aligned} \mathbf {I}_{i}=\mathbf {I}_{G_{i}}+m_{i} \left[ \left( \mathbf {r}_{i}^{\mathrm {T}} \times \mathbf {r}_{i}\right) \mathbf {1}_{3 \times 3}-\left( \mathbf {r}_{i} \times \mathbf {r}_{i}^{\mathrm {T}}\right) \right] . \end{aligned}$$

(89)

Substitution of Eq. (89) into second equation of Eq. (88) leads to:

$$\begin{aligned}&\mathbf {I}_{\mathrm {G}_{i}} \dot{\pmb {\omega }}_{i} = \left( \mathbf {I}_{i}-m_{i}\left[ \left( \mathbf {r}_{i}^{\mathrm {T}} \times \mathbf {r}_{i}\right) \mathbf {1}_{3 \times 3}-\left( \mathbf {r}_{i} \times \mathbf {r}_{i}^{\mathrm {T}}\right) \right] \right) \dot{\pmb {\omega }_{l}} \end{aligned}$$

(90)

$$\begin{aligned}&\pmb {\omega }_{i} \times I_{G_{i}} \pmb {\omega }_{i}=\pmb {\omega }_{i} \times \left( I_{i}-m_{i}\left[ \left( \mathbf {r}_{i}^{\mathrm {T}} \times \mathbf {r}_{i}\right) \mathbf {1}_{3 \times 3}\right. \right. \nonumber \\&\left. \left. \quad -\left( \mathbf {r}_{i} \times \mathbf {r}_{i}^{T}\right) \right] \right) \pmb {\omega }_{i} \end{aligned}$$

(91)

$$\begin{aligned}&\mathbf {r}_{i} \times \dot{\mathbf {v}}_{\mathrm {G}_{i}}=\mathbf {r}_{i} \times \left( \dot{\mathbf {v}}_{i}+\pmb {\omega }_{i} \times \left( \pmb {\omega }_{i} \times \mathbf {r}_{i}\right) +\dot{\pmb {\omega }}_{i} \times \mathbf {r}_{i}\right) \end{aligned}$$

(92)

Moreover, from linear algebra one has:

$$\begin{aligned}&\pmb {a} \times (\pmb {B} \times \pmb {a})=\left( \pmb {a}^{\mathrm {T}} \pmb {a} \; \mathbf {1}_{3 \times 3}-\pmb {a} \; \pmb {a}^{\mathrm {T}}\right) \pmb {B} \end{aligned}$$

(93)

$$\begin{aligned}&\pmb {a} \times (\pmb {B} \times (\pmb {B} \times \pmb {a}))=\pmb {B} \times \left( \pmb {a}^{\mathrm {T}} \pmb {a} \; \mathbf {1}_{3 \times 3}-\pmb {a}\;\pmb {a}^{\mathrm {T}}\right) \pmb {B} \end{aligned}$$

(94)

where $\pmb {a}$ is an arbitrary 3-by-1 vector, and $\pmb {B}$ denotes a 3-by-3 arbitrary matrix.

Substitution of Eqs. (90) to (92) into Eq. (88) and Considering Eqs. (93) and (94), the analogues terms with opposite sign appear in Eq. (88) that by eliminating them one can rewrite Eq. (37) as follows:

$$\begin{aligned} {\mathscr {F}}_{i}^{ \text{ inertia } }=\left[ \begin{array}{c}{m \left( \dot{\mathbf {v}}_{i}+\pmb {\omega }_{i} \times \left( \pmb {\omega }_{i} \times \mathbf {r}_{i}\right) +\dot{\pmb {\omega }}_{i} \times \mathbf {r}_{i}\right) } \\ {\mathbf {I}_{i} \dot{\pmb {\omega }}_{i}+\pmb {\omega }_{i} \times \mathbf {I}_{i} \pmb {\omega }_{i}+m \left( \mathbf {r}_{i} \times \dot{\mathbf {v}}_{i}\right) }\end{array}\right] . \end{aligned}$$

(95)

In this equation, the terms regarding the center of gravity have been eliminated and it is more convenient to use it in the identification process.

1.2 Screw rotation matrices used in this paper

In this Appendix the rotation matrices used in this paper have been illustrated. As discussed in Sect. 3.3, there are three types of coordinate frames, namely, the limb coordinate frames, actuated links’ coordinate frames, and parallelogram links’ coordinate frames. In what follows, the rotation matrices associated with these coordinate frames have been elaborated, afterwards, the Screw rotation matrices have been defined. The coordinate rotation matrices are as follows:

$$\begin{aligned}&{}^B\mathbf {R}_{0i}^{\text {crd}}(\psi )=\left[ \begin{array}{ccc} \cos {\psi } &{} \sin {\psi } &{} 0 \\ -\sin {\psi } &{} \cos {\psi } &{} 0 \\ 0 &{} 0 &{} 1 \end{array}\right] \end{aligned}$$

(96)

$$\begin{aligned}&{}^0\mathbf {R}_{1i}^{\text {crd}}(\theta _{1i})=\left[ \begin{array}{ccc} \cos {\theta _{1i}} &{} 0 &{} -\sin {\theta _{1i}} \\ 0 &{} 1 &{} 0 \\ \sin {\theta _{1i}} &{} 0 &{} \cos {\theta _{1i}} \end{array}\right] \end{aligned}$$

(97)

$$\begin{aligned}&{}^0\mathbf {R}_{2i}^{\text {crd}}(\theta _{2i},\phi _i)= \left[ \begin{array}{ccc} \cos {\phi } &{} \sin {\phi } &{} 0 \\ -\sin {\phi } &{} \cos {\phi } &{} 0 \\ 0 &{} 0 &{} 1 \end{array}\right] \left[ \begin{array}{ccc} \cos {\theta _{2i}} &{} 0 &{} -\sin {\theta _{2i}} \\ 0 &{} 1 &{} 0 \\ \sin {\theta _{2i}} &{} 0 &{} \cos {\theta _{2i}} \end{array}\right] \end{aligned}$$

(98)

According to the coordinate rotation matrices, the Screw rotation matrices have been derived as:

$$\begin{aligned}&{}^B\mathbf {R}_{0i}=\left[ \begin{array}{cc} ^0\mathbf {R}_{1i}^{\text {crd}} &{} \mathbf {0}_{3 \times 3} \\ \mathbf {0}_{3 \times 3} &{} ^0\mathbf {R}_{1i}^{\text {crd}} \end{array}\right] \end{aligned}$$

(99)

$$\begin{aligned}&{}^0\mathbf {R}_{1i}=\left[ \begin{array}{cc} ^0\mathbf {R}_{1i}^{\text {crd}} &{} \mathbf {0}_{3 \times 3} \\ \mathbf {0}_{3 \times 3} &{} ^0\mathbf {R}_{1i}^{\text {crd}} \end{array}\right] \end{aligned}$$

(100)

$$\begin{aligned}&{}^0\mathbf {R}_{2i}= \left[ \begin{array}{cc} ^0\mathbf {R}_{2i}^{\text {crd}} &{} \mathbf {0}_{3 \times 3} \\ \mathbf {0}_{3 \times 3} &{} ^0\mathbf {R}_{2i}^{\text {crd}} \end{array}\right] \end{aligned}$$

(101)

1.3 Statistical Properties of SVD-DLS

In this section, the statistical properties of the proposed SVD-DLS method and classic LS are compared. To the end of investigating the effects of noise, the system output, $\pmb {\tau }_t$, is assumed to be contaminated with additive zero-mean white noise vector, $\mathbf {\eta }_t$, with variance $\mathbf {\sigma }$. The noise vector, denoted with $\mathbf {N}_t$, contains kt independent and identically distributed (iid) random variables. Therefore, the outputs of system can be modeled by:

$$\begin{aligned} {{\pmb {\tau }}_{t}}={\bar{\pmb {\tau }}_{t}}+{{\mathbf {\eta } }_{t}}= \mathbf {D} _{t}^{\mathrm {T}}{{\mathbf {P}}_{t}}+{{\mathbf {\eta } }_{t}} \end{aligned}$$

(102)

where ${\bar{\pmb {\tau }}_{t}}$ denotes the deterministic part of the output. The non-recursive parameter estimation in LS method is calculated as follows:

$$\begin{aligned} {{\hat{\mathbf {P} }}_{t}} = {\left( {\mathbf {X}_{t}}^{\mathrm {T}} \mathbf {X}_{t} \right) }^{-1} {\mathbf {X}_{t}}^{\mathrm {T}} {\mathbf {Y}_{t}} \end{aligned}$$

(103)

where $\mathbf {X}_{t}\in {\mathbb {R}}^{kt\times n}$ shows the augmented matrix of input regressors and is not affected by the additive noise. ${\mathbf {Y}_{t}\in {\mathbb {R}}^{kt}}$ denotes the augmented vector of output measurements. Using the deterministic part of the output (${\bar{\mathbf {Y}}_{t}}$), the estimation of deterministic parameter vector, ${{{\mathbf {P} }}_{t}}={\left( {\mathbf {X}_{t}}^{\mathrm {T}} \mathbf {X}_{t} \right) }^{-1} {\mathbf {X}_{t}}^{\mathrm {T}} {\bar{\mathbf {Y}}_{t}}$, is obtained which is equivalent with the noise-free estimation of parameters. Applying the proposed method leads to the following estimation for the deterministic part of parameters:

$$\begin{aligned} \begin{aligned}&\hat{\mathbf {P}}^{\beta }_t = \underbrace{ {\left( \mathbf {V}_{t}^ \text {p} \pmb {\Lambda } _{t}^ \text {p} \mathbf {V}_{t}^{{ \text {p}^{\mathrm {T}}}} \right) }^{-1} {\mathbf {X}_{t}}^{\mathrm {T}} {\bar{\mathbf {Y}}_{t}} + \mathbf {V}_{\text {nul}} {\pmb {\beta }} } _ {{\mathbf {P}}^{\beta }_t} \\&\quad + {\left( \mathbf {V}_{t}^ \text {p} \pmb {\Lambda } _{t}^ \text {p} \mathbf {V}_{t}^{{ \text {p}^{\mathrm {T}}}} \right) }^{-1} {\mathbf {X}_{t}}^{\mathrm {T}} {{\mathbf {N}}_{t}} \end{aligned} \end{aligned}$$

(104)

The expected value of $\hat{\mathbf {P}}^{\beta }_t$ could be calculated as follows:

$$\begin{aligned} \text {E}\{ \hat{\mathbf {P}}^{\beta }_t |{\mathbf {X}_{t}} \} = {\mathbf {P}}^{\beta }_t + \text {E}\{ {\left( \mathbf {V}_{t}^ \text {p} \pmb {\Lambda } _{t}^ \text {p} \mathbf {V}_{t}^{{ \text {p}^{\mathrm {T}}}} \right) }^{-1} {\mathbf {X}_{t}}^{\mathrm {T}} {{\mathbf {N}}_{t}} \} = {\mathbf {P}}^{\beta }_t \end{aligned}$$

(105)

Since the noise is assumed to be zero-mean and iid, SVD-DLS leads to an unbiased estimation for any ${\pmb {\beta }}$. Thereafter, computing the covariance matrix of the estimated vector leads to:

$$\begin{aligned} \begin{aligned}&\text {cov}\{ \hat{\mathbf {P}}^{\beta }_t |{\mathbf {X}_{t}} \} = {\left( \mathbf {V}_{t}^ \text {p} \pmb {\Lambda } _{t}^ \text {p} \mathbf {V}_{t}^{{ \text {p}^{\mathrm {T}}}} \right) }^{-1} {\mathbf {X}_{t}}^{\mathrm {T}} \text {E}\{ {{\mathbf {N}}_{t}} {{\mathbf {N}}_{t}}^{\mathrm {T}} \} \\&{\mathbf {X}_{t}} {\left( \mathbf {V}_{t}^ \text {p} \pmb {\Lambda } _{t}^ \text {p} \mathbf {V}_{t}^{{ \text {p}^{\mathrm {T}}}} \right) }^{-1} = \mathbf {\sigma } {\left( \mathbf {V}_{t}^ \text {p} \pmb {\Lambda } _{t}^ \text {p} \mathbf {V}_{t}^{{ \text {p}^{\mathrm {T}}}} \right) }^{-1}. \end{aligned} \end{aligned}$$

(106)

It follows from Eq. (66) that ${\mathbf {R}_{t}}= \mathbf {V}_{t}^ \text {p} \pmb {\Lambda } _{t}^ \text {p} \mathbf {V}_{t}^{{ \text {p}^{\mathrm {T}}}}$. Therefore, when invasive measurements are available, the eigenvalues of inverse of $\mathbf {V}_{t}^ \text {p}$ are decreased and as a result, SVD-DLS estimations remain consistent.

A similar procedure can be conducted to show that the covariance matrix of estimated parameters is equal to the Cramer-Rao lower bound. It follows that the proposed method is efficient when the additive noise is white Gaussian. Consequently, the SVD-DLS method has similar facets of the classic LS from the statistical point of view, and in addition, solves problems of estimation windup and insufficient excitations.