2.4 The Dual Number, Dual Vector and Their
2.4.1 Calculus of the Dual Ring
The dual number, first introduced by Clifford in 1873 [9], was further de- veloped by Study in 1901 [10] to represent a dual angle in spatial geometry [11]–[18]. Suppose that the shortest distance between two straight lines in 3D space isdand the angle between their directions isα, then adual anglewas defined by
ˆ
α=α+d
under 2= 0. The geometric meaning for this so-defined dual angle is like a screw motion. Namely, using a screw driver to twist a screw by an angle α while pressing and moving the screw down for a vertical distanced.
Like most common functions in the complex field, a smooth function of a dual number f(a+b) can be expanded into a formal Taylor series at the pointa, and due to2= 0, the expansion will soon become finite:
f(a+b) =f(a) +bf(a) +2b2
2!f(a) +ã ã ã
=f(a) +bf(a), (2.15)
wheref(a) andf(a) are the first and second derivatives, respectively, and evaluated at the pointa. It is interesting that the above general Taylor ex- pansion over the dual ring generates a new dual function. Based on such a compact expansion, it can be easily seen that
ea+b=eaãeb=ea+bea, (2.16)
and also
sin(α+d) = sinα+dcosα (2.17) and
cos(α+d) = cosα−dsinα. (2.18) More significantly, the general expansion in (2.15) can be directly used through the dual-number calculus to determine the first derivative of any smooth function numerically without error and without a symbolical deriva- tion needed by hand. As we have all experienced in numerical computation, to integrate a function numerically in a computer can easily reach at least the forth-order accuracy, such as using the well-known Runge-Kutta algorithm.
However, to calculate the derivative for a function numerically, it is often difficult to reach a satisfactory high-order approximation without a memory of previous sampling data. Now, if a computational software package, such as MATLABT M, can implement an internal subroutine to operate the dual- number calculus, like the complex number calculation that has already been built in MATLABT M, we can calculate numerically any kind of derivatives of a function exactly and instantaneously at a point of interest in a computer via equation (2.15) without going through a symbolical derivation.
As an illustrative example, let
f(x) =xexp(−x2).
We wish to calculate the first derivative off(x) evaluated atx= 3, i.e.,f(3) numerically. Letx= 3 +1 = 3 +. Then,
f(x) = (3+) exp(−(3+)2) = (3+)e−9−6= (3+)e−9(1−6) = (3−17)e−9. Clearly, taking the dual part of the above result yields
Du{f(3 +)}=−17e−9=−2.098×10−3,
which is just the value off(3) because we purposely setb= 1 inx. To verify the numerical result, we may symbolically calculate
f(x) = (1−2x2) exp(−x2).
Then, substitutingx= 3 into the above derivative, we reach the exactly same answer−17e−9=−2.098×10−3.
In the later chapters, we will have to calculate derivatives for much more complicated functions, or high-dimensional vector fields or matrix fields. It can be difficult or even unmanageable to find their symbolical derivatives by hand before programming them for numerical computation. In this case, the above property owned by the dual ring will overwhelmingly demonstrate its unique advantage over any other algorithms for a numerical derivative solution [1].
Furthermore, suppose that we replace a+b in (2.15) by q+q, where˙ q ∈ Rn could be an n-dimensional vector for n = 1,2,ã ã ã and ˙q is its
time-derivative, while the function itselff(ã) could also be a high-dimensional vector or even a matrix (note that in this case, the dimensions in each matrix multiplication must be compatible). Then,
f(q+q) =˙ f(q) +∂f(q)
∂q q˙=f(q) +f˙(q), (2.19) where ˙f(q) =dtdf(q) =∂f(q)∂q q, the time-derivative of the function˙ f(q) based on the derivative chain-rule. This new property in (2.19) will definitely extend the unique advantage of (2.15) to directly finding a time-derivative numeri- cally for a large-scale function, which will be more often required in the later robotics and digital human modeling applications. Of course, the derivative may also be generalized with respect to any parameter other than time, and this will be exhibited more in later developments.
Let us now look at a real example. Suppose that the orientation for a robot link frame is given by the following rotation matrix with respect to the fixed base:
R=
⎛
⎝c1c23 s1 c1c23
s1c23 −c1 s1s23
s23 0 −c23
⎞
⎠,
wheres1= sinθ1,c1= cosθ1,s23= sin(θ2+θ3) andc23= cos(θ2+θ3) as a conventional short notation. At a certain time instant, the three joint angles of the robot are θ1 =π/6, θ2 =π/2 and θ3 = 0 in radians, and the joint velocities are given by ˙θ1 = 0.5, ˙θ2 =−0.4 and ˙θ3 = 0.8 in radians/second.
Then, programming it into MATLABT M, the rotation matrix R will have the following numerical result at the time instant:
>> R =
0.0000 0.5000 0.8660 0.0000 -0.8660 0.5000
1.0000 0 -0.0000
>>
Now, let each joint angle θi be replaced by θi +θ˙i for i = 1,2,3, and taking the dual part out from the final numerical result ˆR=R+S, we will immediately see the output in the MATLABT M working window:
>> S =
-0.3464 0.4330 -0.2500 -0.2000 0.2500 0.4330
0.0000 0 0.4000
>>
Based on (2.19), this matrixSvalue is just the instantaneous time-derivative ofR, i.e.,S= ˙R.
Furthermore, if we wish to determine numerically the partial derivative of R with respect toθ2 at the same time instant, i.e.,S= ∂θ∂R
2, then onlyθ2 in R is replaced by θ2+1 and the other two joint angles remain unchanged.
Clearly, this is equivalent to setting the joint velocity ˙θ2 = 1 and the others θ˙1 = 0 and ˙θ3 = 0 in the above MATLABT M program. The final result is popped out as follows:
>> S =
-0.8660 0 0.0000
-0.5000 0 0.0000
0.0000 0 1.0000
>>
which is just the instantaneous partial derivative ofRwith respect toθ2. Us- ing the same method, we can also calculate the numerical partial derivatives
∂R
∂θ1 and ∂θ∂R
3.