Using a robot arm to reach a specific point with arbitrary orientation
$begingroup$
I want to use a 6-DoF robot arm to reach a user-given point
$$P_{target} = (x, y, z)$$
with arbitrary orientation.
My robot arm is equipped with a laser pen and what I want to do is to aim at point $P_{target}$ along the axis perpendicular to the flange face of my robot arm with a predefined distance $D$.
I want to minimize the movement of my robot arm, that is, with given start pose
$$(x_i, y_i, z_i, q_{xi}, q_{yi}, q_{zi}, q_{wi})$$
and final pose
$$(x_f, y_f, z_f, q_{xf}, q_{yf}, q_{zf}, q_{wf})$$
the value
$${(x_f-x_i)}^2 + {(y_f - y_i)}^2 + {(z_f - z_i)}^2 + {(q_{xf} - q_{xi})}^2 + {(q_{yf} - q_{yi})}^2 + {(q_{zf} - q_{zi})}^2 + {(q_{wf} - q_{wi})}^2$$
can be optimized.
How could I list the equations and get the result
$$(q_{xf}, q_{yf}, q_{zf}, q_{wf})$$
?
Thanks a lot for any hint and help.
optimization robotics
edited Dec 22 '18 at 17:08
timtfj
2,067420
asked Dec 22 '18 at 16:09
Sean LuSean Lu
1
$endgroup$
add a comment |
$begingroup$
I want to use a 6-DoF robot arm to reach a user-given point
$$P_{target} = (x, y, z)$$
with arbitrary orientation.
My robot arm is equipped with a laser pen and what I want to do is to aim at point $P_{target}$ along the axis perpendicular to the flange face of my robot arm with a predefined distance $D$.
I want to minimize the movement of my robot arm, that is, with given start pose
$$(x_i, y_i, z_i, q_{xi}, q_{yi}, q_{zi}, q_{wi})$$
and final pose
$$(x_f, y_f, z_f, q_{xf}, q_{yf}, q_{zf}, q_{wf})$$
the value
$${(x_f-x_i)}^2 + {(y_f - y_i)}^2 + {(z_f - z_i)}^2 + {(q_{xf} - q_{xi})}^2 + {(q_{yf} - q_{yi})}^2 + {(q_{zf} - q_{zi})}^2 + {(q_{wf} - q_{wi})}^2$$
can be optimized.
How could I list the equations and get the result
$$(q_{xf}, q_{yf}, q_{zf}, q_{wf})$$
?
Thanks a lot for any hint and help.
optimization robotics
edited Dec 22 '18 at 17:08
timtfj
2,067420
asked Dec 22 '18 at 16:09
Sean LuSean Lu
1
$endgroup$
$begingroup$
I suspect that what makes the problem difficult is not yet articulated in your Question. On one side you wrote about minimizing "the movement of my robot arm" and give a sum of squares "value" to be "optimized". But it is unclear what choices you have in arguments. The values of the initial and final poses seem to be fixed. Probably you have in mind some steps between these poses. Also there are likely some constraints and interactions between variables that would inform the desired "solution".
$endgroup$
– hardmath
Dec 22 '18 at 16:27
$begingroup$
To make typing equations and formulas less laborious: you only need a dollar sign at the start and end of the whole formula, not round each term. (Or a double dollar sign at each end if you want the formula to appear in "display format" on a line of its own.)
$endgroup$
– timtfj
Dec 22 '18 at 16:42
add a comment |
$begingroup$
I want to use a 6-DoF robot arm to reach a user-given point
$$P_{target} = (x, y, z)$$
with arbitrary orientation.
My robot arm is equipped with a laser pen and what I want to do is to aim at point $P_{target}$ along the axis perpendicular to the flange face of my robot arm with a predefined distance $D$.
I want to minimize the movement of my robot arm, that is, with given start pose
$$(x_i, y_i, z_i, q_{xi}, q_{yi}, q_{zi}, q_{wi})$$
and final pose
$$(x_f, y_f, z_f, q_{xf}, q_{yf}, q_{zf}, q_{wf})$$
the value
$${(x_f-x_i)}^2 + {(y_f - y_i)}^2 + {(z_f - z_i)}^2 + {(q_{xf} - q_{xi})}^2 + {(q_{yf} - q_{yi})}^2 + {(q_{zf} - q_{zi})}^2 + {(q_{wf} - q_{wi})}^2$$
can be optimized.
How could I list the equations and get the result
$$(q_{xf}, q_{yf}, q_{zf}, q_{wf})$$
?
Thanks a lot for any hint and help.
optimization robotics
edited Dec 22 '18 at 17:08
timtfj
2,067420
asked Dec 22 '18 at 16:09
Sean LuSean Lu
1
$endgroup$
I want to use a 6-DoF robot arm to reach a user-given point
$$P_{target} = (x, y, z)$$
with arbitrary orientation.
My robot arm is equipped with a laser pen and what I want to do is to aim at point $P_{target}$ along the axis perpendicular to the flange face of my robot arm with a predefined distance $D$.
I want to minimize the movement of my robot arm, that is, with given start pose
$$(x_i, y_i, z_i, q_{xi}, q_{yi}, q_{zi}, q_{wi})$$
and final pose
$$(x_f, y_f, z_f, q_{xf}, q_{yf}, q_{zf}, q_{wf})$$
the value
$${(x_f-x_i)}^2 + {(y_f - y_i)}^2 + {(z_f - z_i)}^2 + {(q_{xf} - q_{xi})}^2 + {(q_{yf} - q_{yi})}^2 + {(q_{zf} - q_{zi})}^2 + {(q_{wf} - q_{wi})}^2$$
can be optimized.
How could I list the equations and get the result
$$(q_{xf}, q_{yf}, q_{zf}, q_{wf})$$
?
Thanks a lot for any hint and help.
optimization robotics
optimization robotics
edited Dec 22 '18 at 17:08
timtfj
2,067420
asked Dec 22 '18 at 16:09
Sean LuSean Lu
1
edited Dec 22 '18 at 17:08
timtfj
2,067420
asked Dec 22 '18 at 16:09
Sean LuSean Lu
1
edited Dec 22 '18 at 17:08
timtfj
2,067420
edited Dec 22 '18 at 17:08
timtfj
2,067420
edited Dec 22 '18 at 17:08
timtfj
2,067420
2,067420
asked Dec 22 '18 at 16:09
Sean LuSean Lu
1
asked Dec 22 '18 at 16:09
Sean LuSean Lu
1
asked Dec 22 '18 at 16:09
Sean LuSean Lu
1
1
$begingroup$
I suspect that what makes the problem difficult is not yet articulated in your Question. On one side you wrote about minimizing "the movement of my robot arm" and give a sum of squares "value" to be "optimized". But it is unclear what choices you have in arguments. The values of the initial and final poses seem to be fixed. Probably you have in mind some steps between these poses. Also there are likely some constraints and interactions between variables that would inform the desired "solution".
$endgroup$
– hardmath
Dec 22 '18 at 16:27
$begingroup$
To make typing equations and formulas less laborious: you only need a dollar sign at the start and end of the whole formula, not round each term. (Or a double dollar sign at each end if you want the formula to appear in "display format" on a line of its own.)
$endgroup$
– timtfj
Dec 22 '18 at 16:42
add a comment |
$begingroup$
I suspect that what makes the problem difficult is not yet articulated in your Question. On one side you wrote about minimizing "the movement of my robot arm" and give a sum of squares "value" to be "optimized". But it is unclear what choices you have in arguments. The values of the initial and final poses seem to be fixed. Probably you have in mind some steps between these poses. Also there are likely some constraints and interactions between variables that would inform the desired "solution".
$endgroup$
– hardmath
Dec 22 '18 at 16:27
$begingroup$
To make typing equations and formulas less laborious: you only need a dollar sign at the start and end of the whole formula, not round each term. (Or a double dollar sign at each end if you want the formula to appear in "display format" on a line of its own.)
$endgroup$
– timtfj
Dec 22 '18 at 16:42
$begingroup$
I suspect that what makes the problem difficult is not yet articulated in your Question. On one side you wrote about minimizing "the movement of my robot arm" and give a sum of squares "value" to be "optimized". But it is unclear what choices you have in arguments. The values of the initial and final poses seem to be fixed. Probably you have in mind some steps between these poses. Also there are likely some constraints and interactions between variables that would inform the desired "solution".
$endgroup$
– hardmath
Dec 22 '18 at 16:27
$begingroup$
I suspect that what makes the problem difficult is not yet articulated in your Question. On one side you wrote about minimizing "the movement of my robot arm" and give a sum of squares "value" to be "optimized". But it is unclear what choices you have in arguments. The values of the initial and final poses seem to be fixed. Probably you have in mind some steps between these poses. Also there are likely some constraints and interactions between variables that would inform the desired "solution".
$endgroup$
– hardmath
Dec 22 '18 at 16:27
$begingroup$
To make typing equations and formulas less laborious: you only need a dollar sign at the start and end of the whole formula, not round each term. (Or a double dollar sign at each end if you want the formula to appear in "display format" on a line of its own.)
$endgroup$
– timtfj
Dec 22 '18 at 16:42
$begingroup$
To make typing equations and formulas less laborious: you only need a dollar sign at the start and end of the whole formula, not round each term. (Or a double dollar sign at each end if you want the formula to appear in "display format" on a line of its own.)
$endgroup$
– timtfj
Dec 22 '18 at 16:42
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint.
A possible variational formulation assuming for simplicity final position without final orientation.
Supposing you have the end point dynamics as
$$
ddot X = Phi(X, dot X, theta)
$$
where $theta$ are the elbow actuators, you need to obtain
$$
min_{theta}int_0^{t_f} ||dot X|| dt mbox{s. t.} ddot X = Phi(X, dot X,theta), , X(t_f) = X_f
$$
NOTE
We assumed minimum movement as minimum path covered distance.
answered Dec 22 '18 at 17:43
CesareoCesareo
8,6393516
$endgroup$
add a comment |
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$begingroup$
Hint.
A possible variational formulation assuming for simplicity final position without final orientation.
Supposing you have the end point dynamics as
$$
ddot X = Phi(X, dot X, theta)
$$
where $theta$ are the elbow actuators, you need to obtain
$$
min_{theta}int_0^{t_f} ||dot X|| dt mbox{s. t.} ddot X = Phi(X, dot X,theta), , X(t_f) = X_f
$$
NOTE
We assumed minimum movement as minimum path covered distance.
answered Dec 22 '18 at 17:43
CesareoCesareo
8,6393516
$endgroup$
add a comment |
$begingroup$
Hint.
A possible variational formulation assuming for simplicity final position without final orientation.
Supposing you have the end point dynamics as
$$
ddot X = Phi(X, dot X, theta)
$$
where $theta$ are the elbow actuators, you need to obtain
$$
min_{theta}int_0^{t_f} ||dot X|| dt mbox{s. t.} ddot X = Phi(X, dot X,theta), , X(t_f) = X_f
$$
NOTE
We assumed minimum movement as minimum path covered distance.
answered Dec 22 '18 at 17:43
CesareoCesareo
8,6393516
$endgroup$
add a comment |
$begingroup$
Hint.
A possible variational formulation assuming for simplicity final position without final orientation.
Supposing you have the end point dynamics as
$$
ddot X = Phi(X, dot X, theta)
$$
where $theta$ are the elbow actuators, you need to obtain
$$
min_{theta}int_0^{t_f} ||dot X|| dt mbox{s. t.} ddot X = Phi(X, dot X,theta), , X(t_f) = X_f
$$
NOTE
We assumed minimum movement as minimum path covered distance.
answered Dec 22 '18 at 17:43
CesareoCesareo
8,6393516
$endgroup$
Hint.
A possible variational formulation assuming for simplicity final position without final orientation.
Supposing you have the end point dynamics as
$$
ddot X = Phi(X, dot X, theta)
$$
where $theta$ are the elbow actuators, you need to obtain
$$
min_{theta}int_0^{t_f} ||dot X|| dt mbox{s. t.} ddot X = Phi(X, dot X,theta), , X(t_f) = X_f
$$
NOTE
We assumed minimum movement as minimum path covered distance.
answered Dec 22 '18 at 17:43
CesareoCesareo
8,6393516
answered Dec 22 '18 at 17:43
CesareoCesareo
8,6393516
answered Dec 22 '18 at 17:43
CesareoCesareo
8,6393516
answered Dec 22 '18 at 17:43
CesareoCesareo
8,6393516
8,6393516
add a comment |
add a comment |
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$begingroup$
I suspect that what makes the problem difficult is not yet articulated in your Question. On one side you wrote about minimizing "the movement of my robot arm" and give a sum of squares "value" to be "optimized". But it is unclear what choices you have in arguments. The values of the initial and final poses seem to be fixed. Probably you have in mind some steps between these poses. Also there are likely some constraints and interactions between variables that would inform the desired "solution".
$endgroup$
– hardmath
Dec 22 '18 at 16:27
$begingroup$
To make typing equations and formulas less laborious: you only need a dollar sign at the start and end of the whole formula, not round each term. (Or a double dollar sign at each end if you want the formula to appear in "display format" on a line of its own.)
$endgroup$
– timtfj
Dec 22 '18 at 16:42