Suppose I have a 6-DOF robot arm with revolute joints whose tooltip is at some position $\bf{r_0}$ and I want to move it through some trajectory ${\bf r}(t)$.
For simplicity, assume there are just 2 links of length $l_1$ and $l_2$, with orthogonal joints at the ends of each link.
Link masses are $w_1$ and $w_2$, uniformly distributed
The initial joint angles of interest are
$${\bf \theta_0} = \left[\theta_{10}, \theta_{20}, ... \theta_{40}\right]$$
- The maximum available torque from each actuator of interest is
$${\bf T_{max}} = \left[{T_{1}}_{max}, {T_{2}}_{max}, ... {T_{4}}_{max}\right]$$
- The maximum available speed from each actuator of interest is
$${\bf \omega_{max}} = \left[{\omega_{1}}_{max}, {\omega_{2}}_{max}, ... {\omega_{4}}_{max}\right]$$
- The actuators weights are
$${\bf w_a} = \left[w_{a1}, w_{a2}, ... w_{a6}\right]$$
Only the lower 4 actuators are "of interest" because we are interested in the location of the center of mass of the final orthogonal actuators.
How does one go about solving the constrained optimization problem of finding the "optimal" tensor:
$${\bf \theta^*}(t) = \left[{\bf\theta_{1}^*}(t), {\bf\theta_{2}^*}(t), ... {\bf\theta_{5}^*}(t)\right]$$
where we assume that $t$ is discretized appropriately (i.e. not undersampled).
Let's assume optimality in this case means the solution that gets the tooltip through ${\bf r}(t)$ as quickly as possible given the constraints.
