How to get effort values of a robotic joint in N.m? I currently have the force and torque values (in 3D vectors) of the parent and child link (joint wrench) of a revolute joint. My question is how can I get the effort values of a joint? Not in vector form, just the N.m effort value of a given joint.
Let's say I want to know the current effort value of joint j1.
The parent and child links of joint j1 are parent_j1 and child_j1 respectively.
The force and torque vectors of parent_j1 are F=(fx1,fy1,fz1) and T=(tx1,ty1,tz1) respectively.
The force and torque vectors of children_j1 are F=(fx2,fy2,fz2) and T=(tx2,ty2,tz2) respectively.
In robotics this is called the wrench of a joint. Then, how can I get: Effort/Torque(j1) = .... N.m.
Thank you very much!
 A: The effort $Q_i$ of the i-th  joint, described by the (6×1) joint axis twist $\boldsymbol{s}_i$ and known (6×1) joint wrench $\boldsymbol{f}_i$ is
$$ \boxed{ Q_i = \boldsymbol{s}_i \cdot \boldsymbol{f}_i } \tag{1} $$
where $\cdot$ is the appropriate dot product between twists and wrenches, combining like terms (linear with linear, and angular with angular).
The result is torque [Nm] for revolute joints, and force [N] for prismatic joints.

In terms of cartesian vectors (for a revolute joint), if the joint position is $\boldsymbol{r}_i$, the joint axis is $\boldsymbol{z}_i$, the reaction force through the joint is $\boldsymbol{F}_i$ and the reaction moment vector is $\boldsymbol{M}_i$ (summed at the joint position), then the effort is
$$ Q_i = \boldsymbol{F}_i \cdot (\boldsymbol{r}_i \times \boldsymbol{z}_i) + ( \boldsymbol{M}_i + \boldsymbol{r}_i \times \boldsymbol{F}_i) \cdot \boldsymbol{z}_i = \boldsymbol{M}_{i}\cdot\boldsymbol{z}_{i} $$
The above equation (1), is the general form, for any type of 1-DOF joint (revolute, prismatic, screw, etc).
For more complex joints, (1) describes the effort for each DOF.
