# Inertial Parameter Identification in Robotics: A Survey
## 5 The Output Error (OE) and Input Error (IE) Identification Approaches
### Parameter Identification with Hopfield-Tank Recurrent Neural Networks (HTRNN)  
Hopfield-Tank recurrent neural networks are dynamic systems made of a set of \(N\) interconnected units, or _neurons_. Each neuron \(i\) behaves as an integrator coupled with a specific nonlinear _activation function_\(f\) as depicted in Figure 4(b). In its original formulation [87], the neuron continuous state equation is expressed as  
\[\frac{du_{i}(t)}{dt}=\sum_{j=1}^{N}\psi_{ij}s_{j}(t)-\frac{u_{i}(t)}{R_{i}C_{i }}+\kappa_{i}, \tag{51}\]  
where \(u_{i}(t)\) is the internal state of neuron \(i\), \(\kappa_{i}\in\mathbb{R}\) is its input bias, \(\psi_{ij}\in\mathbb{R}\) is the connection weight of neuron \(i\) with neuron \(j\), \(s_{j}(t)=f(u_{j}(t))\in[-1,1]\), and where \(R_{i},C_{i}\in\mathbb{R}_{+}^{*}\) are design parameters corresponding to a resistance and a capacitance respectively. As with [13,14,88] we consider in this work the discrete [PERSON]'s formulation of Hopfield-Tank neural networks (originally described in [89]) since, as explained in [88,90], it is particularly suitable for parametric optimization and parameter identification purpose. In [PERSON]'s formulation, the activation function is a hyperbolic tangent:  
\[s_{j}=f(u_{j})=\alpha_{j}\tanh\left(\frac{u_{j}}{\vartheta_{j}}\right),\ \forall\alpha_{j},\ \vartheta_{j}\in\mathbb{R}_{+}^{*} \tag{52}\]  
This leads to the following discrete neuron state equation:  
\[u_{i}^{k+1}=u_{i}^{k}+\eta\sum_{j=1}^{N}\psi_{ij}\alpha_{j}\tanh(\frac{u_{j}^ {k}}{\vartheta_{j}})+\kappa_{i}\,, \tag{53}\]  
where \(\eta\in\mathbb{R}_{+}^{*}\) is a tuning parameter often referred to as the learning rate of the network. It can be demonstrated (c.f. [88,91]) that such a dynamic system is asymptotically Lyapunov  
Figure 5: Adaline and Hopfield Neural Networks used for Inertial Parameter Identification.  
stable, and that its natural evolution converges toward the minimum of the following energy function:  
\[E = -\frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{N}\psi_{ij}s_{i}s_{j}+\sum_{i=1 }^{N}\kappa_{i}s_{j}, \tag{54a}\] \[= -\frac{1}{2}\boldsymbol{s}^{\top}\boldsymbol{\
abla}\boldsymbol{s} +\boldsymbol{s}^{\top}\boldsymbol{\kappa}, \tag{54b}\]  
where \(\boldsymbol{\
abla}=(\boldsymbol{\
abla}_{ij})_{i,j=1\cdots N}\in\mathbb{R}^ {N\times N}\), \(\boldsymbol{s}=\left[s_{1}\,s_{2}\,\cdots\,s_{N}\right]^{\top}\in\mathbb{R}^{N}\) and \(\boldsymbol{\kappa}=\left[\kappa_{1}\,\kappa_{2}\,\cdots\,\kappa_{N}\right]^ {\top}\in\mathbb{R}^{N}\). Under these conditions, the process of parameter identification can be thought of as matching the L2-norm of the parameter error \(\boldsymbol{\epsilon}\) of Equation (19) with the energy function (54) of the Hopfield network. This can be achieved by selecting:  
\[\begin{cases}\boldsymbol{\
abla}=-\boldsymbol{W}^{\top}\boldsymbol{W}\\ \boldsymbol{\kappa}=-\boldsymbol{W}^{\top}\boldsymbol{y}_{\tau}\\ \boldsymbol{s}=\hat{\beta}_{HTRNN}\end{cases} \tag{55}\]  
The resulting neural network will therefore have as many neurons as base dynamic parameters to identify, and the corresponding parameter estimate \(\hat{\boldsymbol{\beta}}_{HTRNN}\) will converge to the ordinary least-squares (IDIM-OLS) estimate, provided that the appropriate solution range is selected. This range should therefore be carefully adjusted. This is made possible by properly tuning the parameter \(\alpha_{j}\) in (53).