Problem The SpaceFab 845 from Physik Instrumente is a parallel kinematic platform for high accuracy positioning....

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Problem The SpaceFab 845 from Physik Instrumente is a parallel kinematic platform for high accuracy positioning. It has a 3PPSR kinematic structure. This means that each of the 3 legs is a serial chain made of a linear stick- slip piezo-based actuator (along the X axis of the base), then a second stick-slip linear piezo-based actuator (along the Y axis of the base), then a passive spherical joint and finally a passive rotary joint connecting the leg to the moving platform. 1) What is a stick-slip linear piezo-based actuator ? 2) Draw a diagram introducing all the notation needed for modelling the SpaceFab mechanism. 3) Write the expression of the loop-closure equation for the SpaceFab mechanism. 4) Show that, using only the latter without taking into account that the last joint of each leg is a rotary joint, yields an infinite number of solutions for the inverse kinematic model of each leg. What is the shape of this set of solutions ? 5) Show that the last rotary joint in each leg imposes a constraint of the form bu,Tbv=0 with bv=EBXzc where 'u, is the direction of leg i in the base frame, E is the center of the moving platform, B, is the center of the last rotary joint and is the third axis of the reference frame attached to the moving platform (which is orthogonal to the moving platform plane, pointing upward). 6) Write the expression of the complete implicit kinematic model of the SpaceFab. 7) Show that this reduces the number of solutions to the inverse kinematic problem to 2 in the general case. Bonus: provide an algorithm for calculating them, ONLY IF YOU HAVE SOME SPARE TIME ! 8) From the complete implicit kinematic model, show that the implicit differential kinematic model writes as: Vi=1..3, Tb y-u \u{x+qu?u?R[I___[*B]x]t=0 xx+xR[I__[*B+L*u]x] *t=0 Problem The SpaceFab 845 from Physik Instrumente is a parallel kinematic platform for high accuracy positioning. It has a 3PPSR kinematic structure. This means that each of the 3 legs is a serial chain made of a linear stick- slip piezo-based actuator (along the X axis of the base), then a second stick-slip linear piezo-based actuator (along the Y axis of the base), then a passive spherical joint and finally a passive rotary joint connecting the leg to the moving platform. 1) What is a stick-slip linear piezo-based actuator ? 2) Draw a diagram introducing all the notation needed for modelling the SpaceFab mechanism. 3) Write the expression of the loop-closure equation for the SpaceFab mechanism. 4) Show that, using only the latter without taking into account that the last joint of each leg is a rotary joint, yields an infinite number of solutions for the inverse kinematic model of each leg. What is the shape of this set of solutions ? 5) Show that the last rotary joint in each leg imposes a constraint of the form bu,Tbv=0 with bv=EBXzc where 'u, is the direction of leg i in the base frame, E is the center of the moving platform, B, is the center of the last rotary joint and is the third axis of the reference frame attached to the moving platform (which is orthogonal to the moving platform plane, pointing upward). 6) Write the expression of the complete implicit kinematic model of the SpaceFab. 7) Show that this reduces the number of solutions to the inverse kinematic problem to 2 in the general case. Bonus: provide an algorithm for calculating them, ONLY IF YOU HAVE SOME SPARE TIME ! 8) From the complete implicit kinematic model, show that the implicit differential kinematic model writes as: Vi=1..3, Tb y-u \u{x+qu?u?R[I___[*B]x]t=0 xx+xR[I__[*B+L*u]x] *t=0