Although the manipulator joints coordinate fully specify the position and orientation of the manipulator hand, they are not suitable for specifying a goal task because most of the manipulator joint coordinates are not orthogonal and they do not separate position from orientation. In such system, the task is usually specified as sequences of Cartesian knot points through which the manipulator hand or end effector must pass.

4.2.1 Planning straight line Trajectory Using Quaternion

A quaternion is a quadruple numbers, s,a,b and c associated respectively with for units: The real number +1 and three other units i,j and k having cyclical permutation,

Text Box: ij = k jk = I ki = j

ji = -k kj = -I jk = -j

Thus a quaternion Q can written as scalar part s and a vector part v:

Text Box: Q = [S + V] = S + ai + bj + ck = (s,a,b,c) (4.19)

The multiplication of two quaternion can write as

Text Box: Q1Q2 = (S1 + a1i + b1j + c1k) (S2 + a2i + b2i + c2i) Text Box: = (S1S2 – V1.V2 + S2V1 + S1V2 + V1xV2) (4.20)

With the aid of quaternion algebra, finite rotations in space may be dealt with in a simple and efficient manner.

Text Box: θ θ
S = sin ─ and C = cos ─
2 2

A rotation Rot (n,θ) of angle θ about an axis n by a quaternion,

Text Box: θ θ
Rot (n,θ) = cos ─ + sin ─ n (4.21)
2 2

Example: a rotation of 90° about K followed by a rotation of 90° j is represented by quaternion product

Text Box: (cos 45° + j sin 45°) (cos 45° + K sin 45°) = (1/2 + j1/2 + k + i1/2)

Text Box: = cos 60° + sin 60° + i + j + k
─────
√3

Text Box: = Rot i + j + k , 120°
─────
√3

The resultant rotation of 120° about an axis equally inclined to the i, j, k axes.

4.2.1 Cartesian Path Control Scheme

To move the manipulators hand frame along a straight line between two knot points specified by F₀ and F₁ in time T, where each coordinate frame is represented by a homogeneous transformation matrix.

Text Box: Fi = Ri Pi
0 1

Let λ(t) be the remaining fraction of the motion still to be traversed at time t.

Text Box: λ(t) = T - t
─── (4.22)
T1

Where T is the total time needed to traverse the segment and t is time starting from the beginning of the segment traversal.

The total frame position and orientation at given time t,

Text Box: P(t) = P1 – λ(t)(P1 - P0) (4.23)

Text Box: R(t) = R1 Rot [n1 – θλ(t)] (4.24)

If the manipulator hand is required to move from one segment to another while maintaining constant acceleration, then it must accelerate or decelerate from one segment to the next.

From this requirement, the boundary conditions for the segment transition are,

Text Box: P(T1 - τ) = P1 – τΔP1
─── (4.26)
P1

Text Box: P(T1 - τ) = P1 + τΔP2
─── (4.27)
P2