*3.2. Improved GA*

The basic GA is usually employed to optimize the polynomial coefficient or the path length in some problems, where chromosomes are assigned a category of meaning. Referring the composition mechanism of chromosomes studied in [22], an IGA was developed. Chromosomes were assigned three categories of meanings. Each IGA chromosome consists of three parts, as shown in Figure 4, where {*W*1, *W*2, ..., *WN*} denotes the waypoint sequence, {*C*1, *C*2, ..., *CN*} denotes the sequence of joint configurations, and *OPC* denotes a special value corresponding with the cubic coefficients of the arguments in the piecewise-sine functions. Joint configuration *Ck* (*k* = 1, ..., *N*) corresponds with waypoint *Wk*.

Since the time period employed by the space robot for each subpath is positive, value ⎧ −*arcsin*) *<sup>q</sup><sup>i</sup> j f* \* ⎫ 1 3

$$\left\{ \left\{ \frac{2\left[a\_{0j}^{i} - \arcsin\left(\frac{2j}{A}\right)\right]}{a\_{3j}^{i}} \right\} \right\} \text{ in (18) is positive. Therefore, } a\_{3j}^{i} \text{ is expressed as:}$$

$$a\_{3j}^{i} = \text{sign}\left(a\_{0j}^{i} - \arcsin\left(\frac{q\_{ff}^{i}}{A}\right)\right) \cdot abs \left(a\_{3spe}\right) \tag{19}$$

*A* In [24], *a*3*spe* denotes the special value after decoding *OPC*, where *abs*(·) denotes the absolute

value of (·). The encoding mechanism is an essential component of the IGA, which is studied first. Then, the IGA updating mechanism, including initialization, reproduction, crossover, and mutation, is studied. Finally, the IGA work mechanism is developed.

**Figure 4.** Composition of each chromosome in the improved genetic algorithm (IGA).
