Inverse dynamic analysis and trajectory planning for flexible manipulator

Figures & data

Figure 1. Configuration of the flexible manipulator.

Figure 2. The third-order polynomial trajectory for both joints.

Figure 3. The fifth-order polynomial trajectory for both joints.

Figure 4. The LSPB trajectory for both joints.

Table 1. LSPB trajectory's parameters.

θ (t) (rad)
(rad s^−1)	C2 t
(rad s^−2)	C2	0	- C2

Table 2. SM trajectory's parameters.

	First, third, fifth and seventh segments	Second and sixth segments	Fourth segment
Jerk	J(t) = Jmax	J(t) = 0	J(t) = 0
Angular position
Angular velocity
Angular acceleration

Figure 5. The SM trajectory for both joints.

Figure 6. Actuator's torque using the third-order polynomial trajectory (first mode).

Figure 7. Actuator's torque using the third-order polynomial trajectory (second mode).

Figure 8. Actuator's torque using the third-order polynomial trajectory (third mode).

Figure 9. Actuator's torque using the LSPB trajectory (first mode).

Figure 10. Actuator's torque using the LSPB trajectory (second mode).

Figure 11. Actuator's torque using the LSPB trajectory (third mode).

Figure 12. Actuator's torque using SM trajectory (first mode).

Figure 13. Actuator's torque using the SM trajectory (second mode).

Figure 14. Actuator's torque using the SM trajectory (third mode).

Figure 15. Actuator's torque using the fifth-order polynomial trajectory (first mode).

Figure 16. Actuator's torque using the fifth-order polynomial trajectory (second mode).

Figure 17. Actuator's torque using the fifth-order polynomial trajectory (third mode).

Table 3. Robot parameters for simulation.

Parameter	Notation	Numerical value
Length of link 1	L1	0.5 m
Length of link 2	L2	0.5 m
Hub mass	mh	0.1 kg
Hub inertia	Ih	10 × Ibeam
Flexural rigidity	EI	1.0 N m^2
Moment of inertia of link 1	I1	0.0026 kg m^2
Moment of inertia of link 2	I2	0.0026 kg m^2
Mass density per unit length for link 2	ρ	0. 5 kg m^−1
Initial displacement of link 2	x0	5 mm
Nominal final completion time	tf	3.5 s