Enum CoppObjective 

pub enum CoppObjective<'a> {
    Time(f64),
    ThermalEnergy(f64, &'a [f64]),
    TotalVariationTorque(f64, &'a [f64]),
    Linear(f64, &'a [f64], &'a [f64]),
}

Objective terms for COPP optimization. Continuous formulation is shared by COPP2/COPP3; discrete form depends on how b and torque are sampled. Torque notation:

• continuous: $\boldsymbol{\tau}(s)$.
• discrete: tau[i][k] for joint i at s[k].
• COPP2: tau[i][k] is the right-limit value $\boldsymbol{\tau}(s_k^+)$, i.e. computed on interval $[s_k, s_{k+1}]$ from (a[k], a[k+1], b[k]).
• COPP3: tau[i][k] is node value $\boldsymbol{\tau}(s_k)$, computed from (a[k], b[k]).

Variants

Time(f64)

Time objective.

• Continuous: $J_{\mathrm{time}} = w_t\int_{0}^{t_f} 1 \mathrm{d}t = w_t\int_{s_0}^{s_f} \frac{1}{\sqrt{a(s)}} \mathrm{d}s$.
• Discrete (COPP2): J_time = 2*w_t*sum_{k=0}^{n-2} (s[k+1]-s[k])/(sqrt(a[k])+sqrt(a[k+1])).
• Discrete (COPP3): J_time = w_t*sum_k weight_a_time[k]/sqrt(a[k])

ThermalEnergy(f64, &'a [f64])

Thermal-energy objective.

• Continuous: $J_{\mathrm{th}} = w_e\int_{0}^{t_f} \sum_i(\tau_i(s)\nu_i)^2\mathrm{d}t = w_e\int_{s_0}^{s_f} \sum_i(\tau_i(s)\nu_i)^2\frac{1}{\sqrt{a(s)}}\mathrm{d}s$ where nu_i = normalize[i].
• Discrete (COPP2): J_th = 2*w_e*sum_{k=0}^{n-2} (s[k+1]-s[k])/(sqrt(a[k])+sqrt(a[k+1])) * sum_i (tau[i][k]*normalize[i])^2.
• Discrete (COPP3): J_th = w_e*sum_k weight_a_torque[k] * sum_i (tau[i][k]*normalize[i])^2.

TotalVariationTorque(f64, &'a [f64])

Total-variation of torque objective.

• Continuous: $J_{\mathrm{tv}} = w_v\sum_i \int_{0}^{t_f} \left|\frac{d\tau_i}{\mathrm{d}s}(s)\right|\nu_i\mathrm{d}t = w_v\sum_i \int_{s_0}^{s_f} \left|\frac{d\tau_i}{\mathrm{d}s}(s)\right|\nu_i\mathrm{d}s$.
• Discrete: J_tv = w_v*sum_i sum_k |tau[i][k+1]-tau[i][k]|*normalize[i]

Linear(f64, &'a [f64], &'a [f64])

Linear objective over a and b.

• Continuous: $J_{\mathrm{lin}} = w_l\int_{s_0}^{s_f}(\alpha(s)a(s)+\beta(s)b(s))\mathrm{d}s$.
• Discrete (COPP2):
  • alpha.len()==n, beta.len()==n-1
  • J_lin = w_l*( sum_{k=0}^{n-1} alpha[k]*a[k] + sum_{k=0}^{n-2} beta[k]*b[k] )
• Discrete (COPP3):
  • alpha.len()==beta.len()==n
  • J_lin = w_l*sum_{k=0}^{n-1} (alpha[k]*a[k] + beta[k]*b[k])