- #1
diminho
- 3
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Dear all,
I am building an arm with 2 joints(elbow, shoulder) and want to optimally control it to a particular position(wirst). The examples I saw so far(e.g. acrobot) have a target control signal [itex]u(t_f)[/itex] which becomes zero when reaching the target [itex]x_{target}[/itex] in the final time step [itex]t_f[/itex].
In my case, the target control signal is not zero when reaching the target which can be any point in the horizontal plane within the range of the wrist(the arm is hanging down). My idea was to specify the target control as well as the target position which should be reached. So, the cost function
[tex]J=\frac{1}{2}x(t_f)^TS_fx(t_f)+\frac{1}{2}\int^{t_f}_{t_0}(x(t)^TQx(t)+u(t)^TRu(t))dt[/tex]
becomes
[tex]J=\frac{1}{2}(x(t_f)-x_{target}^T)S_f(x(t_f)-x_{target})+\frac{1}{2}\int^{t_f}_{t_0}((x(t)-x_{target})^TQ(x(t)-x_{target})+(u(t)-u_{target})^TR(u(t)-u_{target}))dt[/tex]
(see scholarpedia->optimal control for notation)
As I am building a real system with friction the target control [itex]u_{target}[/itex] does not always lead to the same position. The position where the wrist ends up depends in which state the arm was before the target control is applied.
So, although the wrist is at the correct position it gets penalized for not having the "correct" controls.
Does anyone of you have an idea how to formulate this problem?
PS: x contains the angles of the joints and the angle velocities.
I am building an arm with 2 joints(elbow, shoulder) and want to optimally control it to a particular position(wirst). The examples I saw so far(e.g. acrobot) have a target control signal [itex]u(t_f)[/itex] which becomes zero when reaching the target [itex]x_{target}[/itex] in the final time step [itex]t_f[/itex].
In my case, the target control signal is not zero when reaching the target which can be any point in the horizontal plane within the range of the wrist(the arm is hanging down). My idea was to specify the target control as well as the target position which should be reached. So, the cost function
[tex]J=\frac{1}{2}x(t_f)^TS_fx(t_f)+\frac{1}{2}\int^{t_f}_{t_0}(x(t)^TQx(t)+u(t)^TRu(t))dt[/tex]
becomes
[tex]J=\frac{1}{2}(x(t_f)-x_{target}^T)S_f(x(t_f)-x_{target})+\frac{1}{2}\int^{t_f}_{t_0}((x(t)-x_{target})^TQ(x(t)-x_{target})+(u(t)-u_{target})^TR(u(t)-u_{target}))dt[/tex]
(see scholarpedia->optimal control for notation)
As I am building a real system with friction the target control [itex]u_{target}[/itex] does not always lead to the same position. The position where the wrist ends up depends in which state the arm was before the target control is applied.
So, although the wrist is at the correct position it gets penalized for not having the "correct" controls.
Does anyone of you have an idea how to formulate this problem?
PS: x contains the angles of the joints and the angle velocities.