Table 1 Unactuated screw types in limb 1 and their satisfying conditions

When the pitch of actuated screw is infinitely large, the driving pair is a prismatic pair in the direction of the x-axis

(1) Line vector screws in the same direction as the actuated screw \(\boldsymbol{\$ }_{11}^{1}\). The number of such screws is at least 2. The maximal linear independent number of spatial parallel line vectors is 3, and the maximum number of such screws is 3.

(2) Couple vector screws perpendicular to the actuated screw \(\boldsymbol{\$ }_{11}^{1}\). Such screws cannot be parallel to each other in space. The maximum linearly independent number of plane intersection couple vectors is 2, and the number of such screws is no more than 2.

(3) Line vector screws intersecting with the transmission wrench screw \(\boldsymbol{\$ }_{{{{T}}1}}\).The number of such screws is at most 2. When the number is 2, the screw must intersect with \(\boldsymbol{\$ }_{{{{T}}1}}\) at a point. Such screws have no effect on the output motion of the mechanism, so they are also called inert pairs.

When the actuated screw is a zero-pitch screw, the driving pair is rotated along the y-axis direction or z-axis direction

(1) The number of line vector screws parallel to the x-axis is at least 2 and at most 3. No other line vector screws with different directions can be added between the screws.

(2) The couple vector screws perpendicular to the x-axis. The number of such screws is at most 2.

1) When the number of the screw is 1, if the two ends of the screw are in the same direction as the line vectors connected to it, the axis of the couple vector should be perpendicular to and coplanar with the two end line vectors. If the two ends of the line vector connected with it are orthogonal, the axis direction of the couple vector can only be along the y-axis or z-axis direction, and the couple vector can also be directly connected with the moving platform.

2) When the number of the screw is 2, the two couple vectors are along the y direction and z direction, and at most one couple vector in the same direction as the driving pair is arranged in the 2R parallel sub-chain. When a couple vector is set in the 2R parallel sub-chain, another couple vector is arranged between the 2R parallel sub-chain and the moving platform. Otherwise, two couple vectors are arranged between the 2R parallel sub-chain and the moving platform.

(3) Line vector screws intersecting with the transmission wrench screw \(\boldsymbol{\$ }_{{{{T}}1}}\) and perpendicular to the actuated screw. The number of such screws is at most 2, and they are inert screws.