Attitude dynamic singularities in 3D free-flying manipulators for improved path planning

Abstract

This paper extends the solution, already presented to the trajectory planning problem of 2D free-flying manipulators, to 3D manipulators. It demonstrates it is possible to design a robotic arm with a special dynamic singularity (attitude singular configuration), thus permitting to determine and execute its trajectory without affecting the attitude of the spacecraft carrying it. This methodology provides an exact solution to trajectory planning problems that are usually dealt with by approximate algorithms based on the concept of Disturbance Map. After a theoretical introduction, some educational design examples are presented.

Appendix: Derivation of terms A i

In Sect. 2 we mentioned that the angular momentum of the system composed by the spacecraft and the manipulator, expressed with respect to the global center of mass G, can be written as:

(26)

The aim of this section is to derive the values of terms A _i of Eqs. (2), (3) and (4). The velocity of the ith center of mass is the sum of both the contribution of the spacecraft motion (i.e. the velocity v ₀ of its center of mass and its angular velocity ω ₀) and of the joint motions of links 1÷i

(27)

where vector v _ji is the contribution of the jth joint to the velocity of the center of mass of the ith link assuming $\dot{q}_{j}=1$. In other words $\boldsymbol{v}_{ji}=\partial\dot {\boldsymbol{G}}_{i}/\partial\dot{q}_{j}$.

Vector v _ji can be expressed as:

(28)

The vector term u _j is the direction of the jth joint axis, and p _i is an arbitrary point of the jth joint axis.

The angular velocity of the ith link depends on that of the base, plus the joint velocities of joints preceding it

(29)

where a _j =1 for revolute joints and a _j =0 for prismatic ones.

First of all, we note that system angular momentum does not depend on v ₀; in fact substituting Eq. (26) in Eq. (27) and expanding the term we get

(30)

this condition is true for any vector v ₀. This property is easily proved by the definition of the center of mass

(31)

The angular momentum depends linearly on all joint velocities and so the contribution of each joint can be separately evaluated; therefore the generic term A _i , which represents the angular momentum due to a unitary velocity of ith link, can be calculated by considering null the contribution of the base ω ₀=v ₀=0 and assuming $\dot{q}_{i}=1$ and $\dot{q}_{j}=0$ for all the other joints. In more detail, for revolute joints we obtain:

(32)

and for prismatic joints it is simply:

(33)

Finally the value of A ₀, which is the total inertia tensor, is computed substituting in (26) the linear and angular velocities (27) and (29) evaluated for $\dot{\boldsymbol{Q}}=0$. One gets:

(34)

Thus A ₀ can be written as:

(35)

where $[\underline{\boldsymbol{G}}'_{i} ]^{2}$ denotes a 3×3 skew-symmetric representation of the vector $\boldsymbol {G}'_{i}=\boldsymbol{G}_{i}-\boldsymbol{G}$. For any vector v the “underlined” operator transforms a 3-D vector v into a 3×3 skew-symmetric matrix $[\underline{\boldsymbol{v}} ]$ as seen below:

(36)

The “underlined” operator is useful to express vector cross products (e.g. $\boldsymbol{v}_{1}=\boldsymbol{v}_{2}\times\boldsymbol{v}_{3}= [\underline{\boldsymbol{v}}_{2} ]\boldsymbol{v}_{3}$).

We can split A _i into three sub-terms

(37)

since G _i −G _k =0 if i=k

(38)

(39)

Example, if i=3, n=5, $M=\sum^{n}_{j=0}m_{j}$.

(40)

(41)

if the ith joint is prismatic, remembering the definition of v _ij (Eq. (15)), we recognize that this term does not depend on j and so S _2i =0. Other details are given in [7].