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3.2 Static
In considering static forces in a manipulator, consider each link in the structure and write s force moment balance relationship in terms of the link frames. Finally compute the static torque that acting about the joint axis to be in static equilibrium.
The symbol for the force and torque exerted in a link:
ƒ ί = force exerted by link ί
n ί = torque exerted by link ί
Summing the forces and setting them equal to zero:
ί ƒ ί - ί ƒ ί+1 = 0 (3.12)
Summing the torque and setting them to zero:
ί n ί - ί n ί+1 - ί P ί+1 x ί n ί+1 = 0 (3.13)
By formulate the force moment expression (3.12) and (3.13) this specify the interactions from higher number links to lower number links. The result is:
ί ƒ ί = ί ƒ ί+1 (3.14)
ί n ί = ί n ί+1 + ί P ί+1 x ί ƒ ί+1 (3.15)
To formulate this equations in terms of forces and moments, transform it with rotation matrix describing frame {ί+1} relative to frame {ί}.
ί ƒ ί = ί R ί+1 n ί+1 + ί P ί+1 x ί ƒ ί (3.16)
ί+1
ί n ί = ί R ί+1 n ί+1 + ί P ί+1 x ί ƒ ί (3.17)
ί+1
To find the joint torque, it required maintaining the static equilibrium, the joint axis vector with the moment vector acting on link is compute:
T
Tί = ί n ί ί Ž ί (3.18)
In the case that joint ί is prismatic, this compute the joint actuator force as:
T
Tί = ί ƒ ί ί Ž ί (3.19)
The symbol T is for a linear joint force.