Cable-Driven Robots

Abstract

Cable-driven robots (CDRs) are a special class of parallel mechanisms in which the end-effector is actuated by cables, instead of rigid-linked actuators. They are characterized by lightweight structures with low moving inertia and large workspace, due to the location of the cable winching actuators at the fixed base of the structure, and thereby reducing the mass and inertia of the moving platform. CDRs also possess an intrinsically safe feature due to the cables’ flexibility, which allows CDRs to provide safe manipulation in close proximity to their human counterparts. This chapter will highlight the various research endeavors in the performance analysis of CDRs such as force-closure analysis, stiffness analysis, workspace analysis, and cable tension planning. Several case studies will also be presented to serve as illustrations on the application of the proposed performance analysis tools.

Appendices

Appendix 1: Formulation of the Jacobian and Integration Measure Associated with 2-DOF CDR’s Parameterized Rotations

Jacobian Associated with 2-DOF CDR’s Parameterized Rotations

For the 2-DOF CDR, the time-varying rotation matrix is parameterized as

$$ \mathbf{R}(t)=\mathbf{H}\left({h}_1(t),{h}_2(t)\right)=\mathbf{H}\left(\mathbf{h}(t)\right) $$

(63)

Using chain rule from calculus, Eq. 63 becomes

$$ \dot{\mathbf{R}}=\frac{\partial \mathbf{R}}{\partial {h}_1}{\dot{h}}_1+\frac{\partial \mathbf{R}}{\partial {h}_2}{\dot{h}}_2 $$

(64)

Right multiply Eq. 64 by R ^T and extracting the dual vectors yields

$$ {\omega}^s=\mathbf{Q}\left(\mathbf{H}\left(\mathbf{h}\right)\right)\dot{\mathbf{h}} $$

(65)

where ${\omega}^s=\mathrm{vect}\left(\dot{\mathbf{R}}{\mathbf{R}}^T\right)$ is the angular velocity with respect to the spatial reference frame and $\mathbf{Q}\left(\mathbf{H}\left(\mathbf{h}\right)\right)=\left[\mathrm{vect}\left(\frac{\partial \mathbf{H}}{\partial {h}_1}\right){\mathbf{H}}^T,\mathrm{vect}\left(\frac{\partial \mathbf{H}}{\partial {h}_2}\right){\mathbf{H}}^T\kern0.24em \right]$ is the Jacobian associated with 2-DOF CDR’s parameterized rotations.

For the 2-DOF CDR, H(θ ₁, θ ₂) = R _X(θ ₁)R _Y(θ ₂). Hence,

$$ \begin{array}{l}\mathrm{vect}\left(\frac{\partial \mathbf{H}}{\partial {h}_1}\right){\mathbf{H}}^T={R}_X^{\prime}\left({\theta}_1\right){R}_{\mathrm{Y}}\left({\theta}_2\right){R}_{\mathrm{Y}}\left(-{\theta}_2\right){R}_{\mathrm{X}}\left(-{\theta}_1\right)\\ {}\kern3.72em ={R}_{\mathrm{X}}^{\prime}\left({\theta}_1\right){R}_{\mathrm{X}}\left(-{\theta}_1\right)\end{array} $$

$$ \mathrm{vect}\left(\frac{\partial \mathbf{H}}{\partial {h}_2}\right){\mathbf{H}}^T={R}_{\mathrm{X}}\left({\theta}_1\right){R}_Y^{\prime}\left({\theta}_2\right){R}_{\mathrm{Y}}\left(-{\theta}_2\right){R}_{\mathrm{X}}\left(-{\theta}_1\right) $$

Since vect(R ^′ _i R ^T _i ) = e _i regardless of the value of the parameter and vect(R _i XR ^T _i ) = Rvect(X), the Jacobian is given as

$$ \mathbf{Q}\left(\mathbf{H}\left(\mathbf{h}\right)\right)=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill \cos {\theta}_1\hfill \\ {}\hfill 0\hfill & \hfill \sin {\theta}_1\hfill \end{array}\right] $$

(66)

Integration Measure Associated with 2-DOF CDR’s Parameterized Rotations

Using the XYZ Euler angles parameterization, the rotation matrix is given as

$$ \mathbf{H}\left({\theta}_1,{\theta}_2\right)={\mathbf{R}}_{\mathrm{X}}\left({\theta}_1\right){\mathbf{R}}_{\mathrm{Y}}\left({\theta}_2\right){\mathbf{R}}_{\mathrm{Z}}\left({\theta}_3\right) $$

(67)

Employing the same approach the Jacobian matrix is derived as

$$ \mathbf{Q}\left(\mathbf{H}\left({\theta}_1,{\theta}_2,{\theta}_3\right)\right)=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill \sin {\theta}_2\hfill \\ {}\hfill 0\hfill & \hfill \cos {\theta}_1\hfill & \hfill - \sin {\theta}_1 \cos {\theta}_2\hfill \\ {}\hfill 0\hfill & \hfill \sin {\theta}_1\hfill & \hfill \cos {\theta}_1 \cos {\theta}_2\hfill \end{array}\right] $$

(68)

The determinant of Q(H(θ ₁, θ ₂, θ ₃)) is:

$$ \det \Big(\mathbf{Q}\left(\mathbf{H}\left({\theta}_1,{\theta}_2,{\theta}_3\right)\right)= \cos \left({\theta}_2\right) $$

(69)

Therefore, the integration measure is determined as follows:

$$ \left| \det \Big(\mathbf{Q}\left(\mathbf{H}\left({\theta}_1,{\theta}_2,{\theta}_3\right)\right)\right|=\left| \cos \left({\theta}_2\right)\right| $$

(70)

Appendix 2: Formulation of the Integration Measure for SO(3) Representation in Cylindrical Coordinates

It has been addressed in (Bonev and Gosselin 2005) that the entire rigid body group SO(3) can be visualized as a solid cylinder when the cylindrical coordinates are employed to represent the Tilt-&-Torsion angles, as shown in Fig. 22. However, integration measures need to be introduced when computing the volume of the orientation workspace of rigid body rotations from the T&T angles domain. It can be verified that if Cartesian coordinates are used to represent the T&T angles, the integration measure will be same as that of the Euler angle representation, which is given by sin θ. In this case, the volume of the entire rigid body rotation group is given as

$$ {\displaystyle {\int}_{SO(3)}d{V}_R={\displaystyle {\int}_0^{2\pi }{\displaystyle {\int}_0^{pi}{\displaystyle {\int}_0^{2\pi } \sin \theta d\phi d\theta d\sigma}}}}=8{\pi}^2 $$

(71)

Equation 71 has the same form as when the T&T angles are represented with Cartesian coordinates, i.e., x ≡ ϕ, y ≡ θ, and z ≡ σ. However, since the T&T angles are normally represented with cylindrical coordinates, i.e., x ≡ θ cos ϕ, y ≡ θ sin ϕ, and z ≡ σ, an additional integration measure needs to be included for the change of the coordinate representation. Geometrically, such a transformation of the coordinate representations maps the parametric domains of the T&T angles from a rectangular parallelepiped to a solid cylinder. It can be further verified that determinant of the Jacobian (i.e., the additional integration measure) for the transformation of the coordinate representations is given by $\frac{1}{\theta }$. The resultant integration measure becomes $\left|\frac{ \sin \theta }{\theta}\right|$. It follows that the volume of the entire SO(3) under cylindrical coordinate representation of the T&T angles is given by

$$ {\displaystyle {\int}_{S{O}_{(3)}}d{V}_R={\displaystyle {\int}_{{\mathcal{D}}^2\times \mathcal{R}}\left|\frac{ \sin \theta }{\theta}\right|}\;} dxdydz $$

(72)

where ${\mathcal{D}}^2\times \mathcal{R}$ represents a solid cylinder. If the integration is computed using cylindrical coordinates, Eq. 72 can be rewritten as

$$ {\displaystyle {\int}_{SO(3)}d{V}_R={\displaystyle {\int}_0^{2\pi }{\displaystyle {\int}_0^{pi}{\displaystyle {\int}_0^{2\pi}\left|\frac{ \sin \theta }{\theta}\right|\theta d\phi d\theta d\sigma}}}}=8{\pi}^2 $$

(73)

Although Eqs. 71 and 73 are equivalent for the volume computation of SO(3), they possess different geometrical meanings. Equation 71 is associated with the Cartesian coordinate representation of the T&T angles, while Eq. 73 is associated with the cylindrical coordinate representation of the T&T angles. In Eq. 73, the terms $\left|\frac{ \sin \theta }{\theta}\right|$ and θdϕdθdσ represent the integration measure and the differential volume element, respectively.

Equations 72 and 73 also indicate that the integration measure becomes singular when θ approaches 0 or π. However, with strategic selection of the cylindrical coordinate representation for T&T angles, singularity point at θ = 0 can be avoided. This is a significant feature for the numerical volume computation of SO(3) through its parametric domains.

With the equivalent integration measure derived for the cylindrical coordinate representation of T-&-T angles, the integration or convolution of a rotation-dependent function f (R) over a set of rotations S ∈ SO(3) in the T&T angles domain is given by

$$ {\displaystyle {\int}_{R\in S}f(R)d{V}_R=}\kern0.24em {\displaystyle \int {\displaystyle \int {\displaystyle {\int}_{\phi, \theta, \sigma \in {Q}_t}f\left(R\left\{\phi, \theta, \sigma \right\}\right)}}}\left|\frac{ \sin \theta }{\theta}\right|\theta d\phi d\theta d\sigma $$

(74)

where Q _t denotes the parameter space of the T&T angles (with cylindrical coordinate representation), i.e., a subset of the solid cylinder.

After the finite partition of the solid cylinder, the orientation workspace for a set of rotations S ∈ SO(3) can be numerically computed as

$$ {\displaystyle {\int}_{R\in S}d{V}_R\approx}\kern0.24em {V}_t{\displaystyle \sum \left|\frac{ \sin {\theta}_{i\; jk}}{\theta_{i\; jk}}\right|} $$

(75)

where v _t is the unit volume of the equi-volumetric partition scheme in the cylindrical coordinate representation of the T&T angles. Consequently, Eq. 74 can be written as

$$ {\displaystyle {\int}_{R\in S}f(R)d{V}_R\approx}\kern0.24em {V}_t{\displaystyle \sum f\left(R\left\{{\phi}_{i\; jk},{\theta}_{i\; jk},{\sigma}_{i\; jk}\right\}\right)}\left|\frac{ \sin {\theta}_{i\; jk}}{\theta_{i\; jk}}\right|\kern0.24em $$

(76)

where (ϕ _{i jk} , θ _{i jk} , σ _{i jk} ) ∈ Q _t .