Background
Problem statement
Nomenclature
Methods
Trajectory
Sequence 1
Sequence 2–1
Sequence 2–2
Joint angle calculation
Crawler wheel reference speeds
Sequence 1
Sequence 2–1
Sequence 2–2
Optimization
Parameters
Objective function
Constraints
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Maximum stretch This linear constraint imposes a maximum length the power shovel can stretch its arm, where D is a power shovel specific constant.Equation (21) is obtained by adding \(P1 + P2 \le D\) and \(P3 + P4 \le D\) inequalities for the two stages.$$\begin{aligned} P1 + P2 + P3 + P4 \le 2D \end{aligned}$$(21)
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Singular configurations Once the bucket position (P2 or P3) is decided or fixed, the power shovel should have a valid joint configuration at any given time instance in the trajectory to reach it. Based on intersection of circles having radii equal to effective boom length (Fig. 6) and arm length, the following two nonlinear constraints can be imposed:$$\begin{aligned} (x_1-x_2)^2+(z_1-z_2)^2\le (L_1+L_2)^2 \end{aligned}$$(22)where \(x_1\), \(x_2\), \(z_1\), \(z_2\), \(L_1\) and \(L_2\) are described as in Joint Angle Calculation section. \(\left. x_i\right| _{i=1,2}\) and \(\left. z_i\right| _{i=1,2}\) are derived from \(\left. Pi\right| _{i=1,2,3,4}\) and \(\alpha _F\).$$\begin{aligned} (x_1-x_2)^2+(z_1-z_2)^2\ge (L_1-L_2)^2 \end{aligned}$$(23)
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Collision The power shovel must not collide with the step-like structure throughout the obstacle surmount operation. Since the end effector (bucket) is fixed, collision is checked between the step-like structure and the crawler of the power shovel.
Results
General | Specific | Value |
---|---|---|
Mass | Crawler mass | 4.24 kg |
Platform mass | 4.53 kg | |
Boom mass | 0.9 kg | |
Arm mass | 0.29 kg | |
Bucket mass | 0.47 kg | |
Length | Boom effective length | 0.359 m |
Arm length | 0.171 m | |
Bucket length | 0.11 m | |
Crawler wheel radius | 0.033 m | |
Step height | 0.066 m |
Parameter | Value |
---|---|
Time step | 0.002 s |
Friction coefficient | 0.7 |
Algorithm | Dantzig’s (dWorldStep) |
Constraint force mixing (cfm) | 1e−8 |
Error reduction (erp) | 0.2 |
Exhaustive fast batch simulations
SQP optimization
Parameter | Value |
---|---|
Free variables | 4 |
Objective functions | 1 |
Iterations max. | 500 |
Final norm (\(\epsilon\)) | 1.0e−8 |
Perturbation size (\(\lambda\)) | 0.01 |