Application Area 5: Multi-objective Optimization

Systems are required to satisfy multiple objectives. These problems require a slightly different approach.

Multi-objective optimization may be stated as follows:

Minimize	$f = {[ψ_{01} (x, b), ψ_{02} (x, b), \dots, ψ_{0 n} (x, b)]}^{T}$ $f = {[ψ_{01} (x, b), ψ_{02} (x, b), \dots, ψ_{0 n} (x, b)]}^{T}$	(objective function)
Subject to	$ψ_{i} (x, b) \geq 0, i = 1, \dots, p$ $ψ_{i} (x, b) \geq 0, i = 1, \dots, p$	(inequality constraints)
	$ψ_{i} (x, b) = 0, i = p + 1, \dots, m$ $ψ_{i} (x, b) = 0, i = p + 1, \dots, m$ $b_{L} \leq b \leq b_{U}$ $b_{L} \leq b \leq b_{U}$	(equality constraints) (design limits)

The functions $ψ (x, b)$ $ψ (x, b)$ in the cost and constraint functions are assumed to have the form:

$ψ (x, b) = ψ_{0} (x, b) + T_{f} \int T_{0} L (x, b) d t$ $ψ (x, b) = ψ_{0} (x, b) + \int_{T_{0}}^{T_{f}} L (x, b) d t$

In MotionSolve, a multi-objective problem is transformed to a single-objective problem by redefining the cost function as follows:

Minimize:	$f = \sum_{j = 0}^{n} w_{j} ψ_{0 j} (x, b)$ $f = \sum_{j = 0}^{n} w_{j} ψ_{0 j} (x, b)$	(objective function)
Subject to:	$ψ_{i} (x, b) \geq 0, i = 1, \dots, p$ $ψ_{i} (x, b) \geq 0, i = 1, \dots, p$ $ψ_{i} (x, b) = 0, i = p + 1, \dots, m$ $ψ_{i} (x, b) = 0, i = p + 1, \dots, m$ $b_{L} \leq b \leq b_{U}$ $b_{L} \leq b \leq b_{U}$	(inequality constraints) (equality constraints) (design limits)

Minimize:

f = \sum_{j = 0}^{n} w_{j} ψ_{0 j} (x, b)

$f = \sum_{j = 0}^{n} w_{j} ψ_{0 j} (x, b)$

(objective function)

Subject to:

ψ_{i} (x, b) \geq 0, i = 1, \dots, p

$ψ_{i} (x, b) \geq 0, i = 1, \dots, p$

$ψ_{i} (x, b) = 0, i = p + 1, \dots, m$ $ψ_{i} (x, b) = 0, i = p + 1, \dots, m$

$b_{L} \leq b \leq b_{U}$ $b_{L} \leq b \leq b_{U}$

(inequality constraints)

(equality constraints)

(design limits)

$w_{j} > 0$ $w_{j} > 0$ are a set f weights that reflect the relative importance of the various objectives.

Example 1

Design Goal: Considering a kinematic SLA suspension, the design goal is to modify hard points in a suspension to obtain desired ride behavior.

The ride behavior is characterized by six objectives:

Ride Steer @ Design = -4°/m
Ride Steer @ Full Jounce = -4°/m
Ride Camber @ Design = +5°/m
Ride Camber @ Full Jounce = +5°/m
Ride Caster @ Design = -8°/m
Ride Caster @ Full Jounce = -8°/m

Ride Steer = $\frac{\partial s t e e r A n g l e}{\partial r i d e H e i g h t}$ $\frac{\partial s t e e r A n g l e}{\partial r i d e H e i g h t}$

Ride Camber = $\frac{\partial s t e e r A n g l e}{\partial r i d e H e i g h t}$ $\frac{\partial s t e e r A n g l e}{\partial r i d e H e i g h t}$

Ride Caster = $\frac{\partial c a s t e r A n g l e}{\partial r i d e H e i g h t}$ $\frac{\partial c a s t e r A n g l e}{\partial r i d e H e i g h t}$

Design Variables

Various hard point locations in the suspension are available as design variables. For this problem, there are 26 design variables.

Upper Control Arm Inboard hard-points (Fore/Aft) XYZ	(6 DV)
Lower Control Arm Inboard hard-points (Fore/Aft) XYZ	(6 DV)
Upper Ball Joint hard-point XYZ	(3 DV)
Lower Ball Joint hard-point XYZ	(3 DV)
Tie rod Inner & Outer XYZ	(6 DV)
Spindle Alignment Point Location XZ	(3 DV)

Results

For this example, the optimization method worked quite well.

Response	Target	Actual
Ride Steer @ Design	-4 °/m	-4.107 °/m
Ride Steer @ Full Jounce	-4 °/m	-3.898 °/m
Ride Camber @ Design	+5 °/m	+4.937 °/m
Ride Camber @ Full Jounce	+5 °/m	+5.024 °/m
Ride Caster @ Design	-8 °/m	-8.402 °/m
Ride Caster @ Full Jounce	-8 °/m	-7.594 °/m

Example 2

This is another suspension example for a bus. The task at hand is to design the steering system to meet required metrics. In a properly design steering system:

The left and right wheels have zero steer when the steering wheel angle is zero.
The steering angle difference between the left and right wheels should be zero at maximum steer and minimum steer.
Steering ratio refers to the ratio between the turn of the steering wheel and the turn of the wheels. The steering ratio at zero steering wheel angle is required to be 9.27.

The problem has four objectives:

Left Wheel Steer Angle @ zero Steering Wheel Angle =0
Right Steer Angle @ zero Steering Wheel Angle =0
Steer Angle Difference between left and right wheels = 0
Steering Ratio @ Zero Steering Wheel Angle = 9.27

Design Variables

Drag Link Ball Position X-Y-Z Coordinates
Pitman Arm Ball Position X-Y-Z Coordinates

Results

For this example, the optimization method worked quite well. The results obtained are shown below:

Response	Target	Actual
Left Steer @ Zero SWA	0°	+0.001°
Right Steer @ Zero SWA	0°	+0.000°
Steering difference (L vs. R)	0°	-0.003°
Steering ratio @ Zero SWA	9.27	9.271