A possible correlation between the basal ganglia motor function and the inverse kinematics calculation

Abstract

The main hypothesis of this study, based on experimental data showing the relations between the BG activities and kinematic variables, is that BG are involved in computing inverse kinematics (IK) as a part of planning and decision-making. Indeed, it is assumed that based on the desired kinematic variables (such as velocity) of a limb in the workspace, angular kinematic variables in the joint configuration space are calculated. Therefore, in this paper, a system-level computational model of BG is proposed based on geometrical rules, which is able to compute IK. Next, the functionality of each part in the presented model is interpreted as a function of a nucleus or a pathway of BG. Moreover, to overcome existing redundancy in possible trajectories, an optimization problem minimizing energy consumption is defined and solved to select an optimal movement trajectory among an infinite number of possible ones. The validity of the model is checked by simulating it to control a three-segment manipulator with rotational joints in a plane. The performance of the model is studied for different types of movement including different reaching movements, a continuous circular movement and a sequence of tracking movements. Furthermore, to demonstrate the physiological similarity of the presented model to the BG structure, the neuronal activity of each part of the model considered as a BG nucleus is verified. Some changes in model parameters, inspired by the dopamine deficiency, also allow simulating some symptoms of Parkinson’s disease such as bradykinesia and akinesia.

Appendices

Appendix A: Deriving the inverse kinematic function

The forward function f _A , a part of the forward kinematic function is defined in Eq. (24) (see Eq. (5)):

$$ f_{A}: \vec{v}_{i} = \vec{\dot \phi}_{i} \times \vec{a}_{i} $$

(24)

By cross-production of $\vec {a}_{i}$ in both sides of the Eq. (24) we have:

$$ \vec{a}_{i} \times \vec{v}_{i} = \vec{a}_{i} \times \vec{\dot \phi}_{i} \times \vec{a}_{i} $$

(25)

According to the vector algebra, for three arbitrary vectors $\vec {u}$, $\vec {v}$ and $\vec {w}$, the following general relation between the cross-product and dot-product is satisfied (Santos & Musaev 2014):

$$ \vec{u} \times \vec{v} \times \vec{w} = \left( \vec{u}.\vec{w}\right)\vec{v} - \left( \vec{u}.\vec{v}\right)\vec{w} $$

(26)

Therefore, based on Eq. (26), the Eq. (25) could be rewritten in Eq. (27):

$$ \vec{a}_{i} \times \vec{v}_{i} = \left( \vec{a}_{i}.\vec{a}_{i}\right)\vec{\dot \phi}_{i} - \left( \vec{a}_{i}.\vec{\dot \phi}_{i}\right)\vec{a}_{i} $$

(27)

The problem is considered for a 2-D horizontal plane, therefore we have:

$$ \begin{array}{llll} & \vec{a}_{i} \bot \vec{\dot \theta}_{j}; && j = 1, 2, \cdots, n \\ \Rightarrow & \vec{a}_{i} \bot \vec{\dot \phi}_{j}; && j = 1, 2, \cdots, n \\ \Rightarrow & \vec{a}_{i}.\vec{\dot \phi}_{j} &= 0; & j = 1, 2, \cdots, n \end{array} $$

(28)

Now, since, $\vec {a}_{i}.\vec {a}_{i} = ||\vec {a}_{i}||^{2}$, Eq. (27) is rewritten based on Eq. (28) in Eq. (29):

$$ \begin{array}{lll} & \vec{a}_{i} &\times \vec{v}_{i} = ||\vec{a}_{i}||^{2}\vec{\dot \phi}_{i} \\ \Rightarrow & \vec{\dot \phi}_{i} & = \frac{\vec{a}_{i} \times \vec{v}_{i}}{||\vec{a}_{i}||^{2}} \end{array} $$

(29)

Based on Eq. (6), $\vec {\dot \phi }_{i} = {\sum }_{j=1}^{i}\vec {\dot \theta }_{j}$, thus Eq. (29) is rewritten in Eq. (30):

$$ \vec{\dot \theta}_{i} = \frac{\vec{a}_{i} \times \vec{v}_{i}}{||\vec{a}_{i}||^{2}} - \vec{\dot \phi}_{i-1} $$

(30)

Appendix B: Solving the optimization problem for a three-segment manipulator in a 2-D plane

In order to solve the optimization problem $\mathcal {P}$ (Eq. (17)), based on Lagrange’s method, the augmented objective function ($\mathcal {L}$) is rewritten in Eq. (31), considering the constraints:

$$ \begin{array}{lll} \mathcal{L} & = & \frac{1}{2}\left\{w_{1}({\Gamma}_{1}\vec{v}_{3,o}^{D})^{T}({\Gamma}_{1}\vec{v}_{3,o}^{D}) + w_{2}({\Gamma}_{2}\vec{v}_{3,o}^{D})^{T}({\Gamma}_{2}\vec{v}_{3,o}^{D})\right.\\ \\ & & \left.+ w_{3}\left\{(I - {\Gamma}_{1}-{\Gamma}2)\vec{v}_{3,o}^{D}\right\}^{T}\left\{(I - {\Gamma}_{1}-{\Gamma}2)\vec{v}_{3,o}^{D}\right\} \right\} \\ \\ & & + \lambda_{1}\vec{a}_{1}^{T}({\Gamma}_{1}\vec{v}_{3,o}^{D}) + \lambda_{2}\vec{a}_{2}^{T}({\Gamma}_{2}\vec{v}_{3,o}^{D}) \\ \\ & & + \lambda_{3}\vec{a}_{3}^{T}\left( I-{\Gamma}_{1}\vec{v}_{3,o}^{D}-{\Gamma}_{2}\vec{v}_{3,o}^{D}\right) \end{array} $$

(31)

where T is the notation of the transpose of a matrix or vector and λ _i (i =1, 2, 3 is a Lagrange multiplier. For simplicity the following notation is defined:

$$\vec{v} \equiv \vec{v}_{3,o}^{D}$$

Based on the necessary condition for optimality (Boyd & Vandenberghe 2004), the following equations should be satisfied for the optimum values of two matrices, Γ₁ and Γ₂:

$$ \left\{ \begin{array}{l} \frac{\partial \mathcal{L}}{\partial {\Gamma}_{1}} = 0 \\ \\ \frac{\partial \mathcal{L}}{\partial {\Gamma}_{2}} = 0 \end{array}\right. $$

(32)

Generally for arbitrary vector $\vec {x}$ and arbitrary matrix B, the following relation is always true (Fukunaga 1990; Searle 2006):

$$ \vec{x}^{T}B^{T}B\vec{x} = tr\{{B^{T}B\vec{x}\vec{x}^{T}}\} $$

(33)

and for two arbitrary vectors x ₁ and x ₂ and the arbitrary matrix B the following relation is always true (Fukunaga 1990; Searle 2006):

$$ \vec{x}_{1}^{T}B\vec{x_{2}} = tr\{B\vec{x_{2}}\vec{x_{1}}^{T}\} $$

(34)

where tr is the notation of trace of a matrix (the sum of elements of the main diagonal of a matrix).

To calculate the derivatives in the Eq. (32), the augmented objective function is written in Eq. (35):

$$ \begin{array}{lll} \mathcal{L} & = & \frac{1}{2}\left\{w_{1}tr\left\{{{\Gamma}_{1}^{T}}{\Gamma}_{1}\vec{v}\vec{v}^{T}\right\} + w_{2}tr\left\{{{\Gamma}_{2}^{T}}{\Gamma}_{2}\vec{v}\vec{v}^{T}\right\}\right.\\ \\ && \left.+ w_{3}tr\left\{(I-{\Gamma}_{1}-{\Gamma}_{2})^{T}(I-{\Gamma}_{1}-{\Gamma}_{2})\vec{v}\vec{v}^{T}\right\}\right\} \\ \\ && + \lambda_{1}tr\left\{{\Gamma}_{1}\vec{v}\vec{a}_{1}^{T}\right\} + \lambda_{2}tr\left\{{\Gamma}_{2}\vec{v}\vec{a}_{2}^{T}\right\} \\ \\ && +\lambda_{3}tr\left\{(I-{\Gamma}_{1}-{\Gamma}_{2})\vec{v}\vec{a}_{3}^{T}\right\} \end{array} $$

(35)

From the first equation, the following equation is derived (Fukunaga 1990; Fang & Zhang 1990):

$$ \begin{array}{ll} &\frac{\partial \mathcal{L}}{\partial {\Gamma}_{1}} = 0 \\ \\ \Rightarrow &w_{1}\vec{v}\vec{v}^{T}{{\Gamma}_{1}^{T}} \\ \\ &- w_{3}\vec{v}\vec{v}^{T}\left( I-{{\Gamma}_{1}^{T}}-{{\Gamma}_{2}^{T}}\right)\\ \\ &= \lambda_{3}\vec{v}\vec{a}_{3}^{T} - \lambda_{1}\vec{v}\vec{a}_{1}^{T} \\ \\ \Rightarrow &(w_{1}+w_{3})\vec{v}\vec{v}^{T}{{\Gamma}_{1}^{T}} \\ &+ w_{3}\vec{v}\vec{v}^{T}{{\Gamma}_{2}^{T}}\\ \\&= \lambda_{3}\vec{v}\vec{a}_{3}^{T} - \lambda_{1}\vec{v}\vec{a}_{1}^{T} + w_{3}\vec{v}\vec{v}^{T} \end{array} $$

(36)

and from the second one, the following equation is obtained:

$$ \begin{array}{ll} &\frac{\partial \mathcal{L}}{\partial {\Gamma}_{2}} = 0 \\ \\ \Rightarrow &w_{3}\vec{v}\vec{v}^{T}{{\Gamma}_{1}^{T}} \\ \\ &+ (w_{1}+w_{3})\vec{v}\vec{v}^{T}{{\Gamma}_{2}^{T}}\\ \\ &= \lambda_{3}\vec{v}\vec{a}_{3}^{T} - \lambda_{2}\vec{v}\vec{a}_{2}^{T} + w_{3}\vec{v}\vec{v}^{T} \end{array} $$

(37)

For simplicity, the following auxiliary variables are defined:

$$ \left\{ \begin{array}{ll} \rho_{1} &= w_{2} + w_{3} \\ \\ \rho_{2} &= w_{1} + w_{3} \\ \\ \rho_{3} &= w_{1} + w_{2} \\ \\ V^{-1} &= (\vec{v}\vec{v}^{T})^{-1} \end{array}\right. $$

(38)

By multiplying V ⁻¹ from left to both sides of Eqs. (36) and (37), the following system of linear equations is obtained:

$$ \left\{ \begin{array}{lllll} \rho_{2}{{\Gamma}_{1}^{T}} &+& w_{3}{{\Gamma}_{2}^{T}} &=& \lambda_{3}V^{-1}\vec{v}\vec{a}_{3}^{T} - \lambda_{1}V^{-1}\vec{v}\vec{a}_{1}^{T} + w_{3}I \\ \\ w_{3}{{\Gamma}_{1}^{T}} &+& \rho_{1}{{\Gamma}_{2}^{T}} &=& \lambda_{3}V^{-1}\vec{v}\vec{a}_{3}^{T} - \lambda_{2}V^{-1}\vec{v}\vec{a}_{2}^{T} + w_{3}I \end{array} \right. $$

(39)

By solving this system of equations, the following solutions are derived:

$$ \left\{ \begin{array}{lll} {\Gamma}_{1} &=& \frac{1}{w}\left( w_{2}\lambda_{3}V^{-1}\vec{v}\vec{a}_{3}^{T} - \rho_{1}\lambda_{1}V^{-1}\vec{v}\vec{a}_{1}^{T} \right. \\ && \left.+ w_{3}\lambda_{2}V^{-1}\vec{v}\vec{a}_{2}^{T} + w_{2}w_{3}I\right) \\ \\ {\Gamma}_{2} &=& \frac{1}{w}\left( w_{1}\lambda_{3}V^{-1}\vec{v}\vec{a}_{3}^{T} - \rho_{2}\lambda_{2}V^{-1}\vec{v}\vec{a}_{2}^{T} \right. \\ && \left.+ w_{3}\lambda_{1}V^{-1}\vec{v}\vec{a}_{1}^{T} + w_{1}w_{3}I\right) \\ \\ {\Gamma}_{3} &=& \frac{1}{w}\left( -\rho_{3}\lambda_{3}V^{-1}\vec{v}\vec{a}_{3}^{T} + w_{2}\lambda_{1}V^{-1}\vec{v}\vec{a}_{1}^{T} \right. \\ && \left.+ w_{1}\lambda_{2}V^{-1}\vec{v}\vec{a}_{2}^{T} + w_{2}w_{1}I\right) \end{array} \right. $$

(40)

where, w is an auxiliary variable defined in Eq. (41):

$$ w = w_{1}w_{2} + w_{1}w_{3} + w_{2}w_{3} $$

(41)

Now, to calculate the values of the Lagrange multipliers, the relations in Eq. (40) are replaced in constraints in the optimization problem $\mathcal {P}$ (Eq. (17)) which leads to the following system of linear equations:

$$ \begin{array}{ll} &\left\{\begin{array}{lll} {\Gamma}_{1}\vec{v}\vec{a}_{1}^{T} &=& 0 \\ \\ {\Gamma}_{2}\vec{v}\vec{a}_{2}^{T} &=& 0 \\ \\ {\Gamma}_{3}\vec{v}\vec{a}_{3}^{T} &=& 0 \end{array} \right. \\ \\ \Rightarrow &\left\{ \begin{array}{lllllll} \rho_{1}b_{11}\lambda_{1} &-& w_{3}b_{12}\lambda_{2} &-& w_{2}b_{13}\lambda_{3} &=& \alpha_{1} \\ \\ -w_{3}b_{12}\lambda_{1} &+& \rho_{2}b_{22}\lambda_{2} &-& w_{1}b_{23}\lambda_{3} &=& \alpha_{2} \\ \\ -w_{2}b_{13}\lambda_{1} &-& w_{1}b_{23}\lambda_{2} &+& \rho_{3}b_{33}\lambda_{3} &=& \alpha_{3} \end{array} \right. \end{array} $$

(42)

where, b _{i j} and α _i (i,j = 1,2,3) are defined in Eq. (43):

$$ \left\{ \begin{array}{lll} b_{ij} &= \vec{a}_{i}^{T}\vec{a}_{j} & i,j = 1,2,3\\ \\ d &= \vec{v}^{T}(\vec{v}\vec{v}^{T})^{-1}\vec{v} &\\ \\ c_{i} &= \vec{v}^{T}\vec{a}_{i} & i = 1,2,3 \\ \\ \alpha_{i} &= w_{j}w_{k}\frac{c_{i}}{d} & i,j,k = 1,2,3 (i \neq j \neq k) \end{array} \right. $$

(43)

By solving the system of equations defined in Eq. (42), the proper values for Lagrange multipliers are derived in Eq. (44):

$$ \left\{ \begin{array}{ll} \lambda_{1} &= \frac{\zeta_{1}\mu_{2}+\zeta_{2}\beta}{\mu_{1}\mu_{2}-\beta^{2}} \\ \\ \lambda_{2} &= \frac{\zeta_{2}\mu_{1}+\zeta_{1}\beta}{\mu_{1}\mu_{2}-\beta^{2}} \\ \\ \lambda_{3} &= \frac{\alpha_{3}(\mu_{1}\mu_{2}-\beta^{2}) + w_{2}b_{13}(\zeta_{1}\mu_{2}+\zeta_{2}\beta)+w_{1}b_{23}(\zeta_{2}\mu_{1}+\zeta_{1}\beta)}{(\mu_{1}\mu_{2}-\beta^{2})(\rho_3b_{33})} \end{array} \right. $$

(44)

where the auxiliary variables are defined in Eq. (45) based on the known values:

$$ \left\{ \begin{array}{ll} \beta &= w_{3}\rho_{3}b_{12}b_{33} + w_{1}w_{2}b_{13}b_{23} \\ \\ \mu_{1} &= \rho_{1}\rho_{3}b_{11}b_{33} - (w_{2}b_{13})^{2} \\ \\ \mu_{2} &= \rho_{2}\rho_{3}b_{22}b_{33} - (w_{1}b_{23})^{2} \\ \\ \zeta_{1} &= \alpha_{1}\rho_{3}b_{33} + w_{2}b_{13}\alpha_{3} \\ \\ \zeta_{2} &= \alpha_{2}\rho_{3}b_{33} + w_{1}b_{23}\alpha_{3} \end{array} \right. $$

(45)

by replacing the derived values for λ ₁, λ ₂ and λ ₃ in Eq. (40), proper matrices to obtain proper linear velocities are calculated.

Appendix C: Analysis of the effect of STN

Based on definition of S ₂ in Eq. (19), Eq. (21) could be extended as Eq. (46):

$$ \begin{array}{lll} \vec{\dot\tau}_{i}(t) &= &+stn_{i}(t) - stn_{i}(t-1) \\ \\ & = &-gpe_{i}(t-1) + \vec{\dot \phi}_{i}^{E}(t-\delta) \\ \\ &&+ gpe_{i}(t-2) - \vec{\dot \phi}_{i}^{E}(t-\delta-1) \\ \\ &=& - \vec{\dot \phi}_{i-1}^{E}(t-\delta-1) -stn_{i}(t-2) \\ \\ && + \vec{\dot \phi}_{i}^{E}(t-\delta) + stn_{i}(t-3) \\ \\ &&+ \vec{\dot \phi}_{i-1}^{E}(t-\delta-1)- \vec{\dot \phi}_{i}^{E}(t-\delta-2) \\ \\ &=& +{\Delta}\vec{\dot \phi}_{i}^{E}(t-\delta) - {\Delta}\vec{\dot \phi}_{i-1}^{E}(t-\delta-1) - \vec{\dot\tau}_{i}(t-2) \\ \\ &=& +{\Delta}\vec{\dot \phi}_{i}^{E}(t-\delta) - {\Delta}\vec{\dot \phi}_{i-1}^{E}(t-\delta-1) \\ \\ && -{\Delta}\vec{\dot \phi}_{i}^{E}(t-\delta-2) + {\Delta}\vec{\dot \phi}_{i-1}^{E}(t-\delta-3) + \vec{\dot\tau}_{i}(t-4) \\ \\ &=& +\nabla\vec{\dot \phi}_{i}^{E}(t-\delta) - \nabla\vec{\dot \phi}_{i-1}^{E}(t-\delta-1) + \vec{\dot\tau}_{i}(t-4) \\ \\ &=& +\nabla\vec{\dot \phi}_{i}^{E}(t-\delta) - \nabla\vec{\dot \phi}_{i-1}^{E}(t-\delta-1) \\ \\ && +\nabla\vec{\dot \phi}_{i}^{E}(t-\delta-4) - \nabla\vec{\dot \phi}_{i-1}^{E}(t-\delta-5) + \vec{\dot\tau}_{i}(t-8) \\ \\ & \approx & +{\sum}_{d=0}^{T}\left( \nabla\vec{\dot \phi}_{i}^{E}(t-\delta-4 \times d)\right.\\ \\ && \left.-\nabla\vec{\dot \phi}_{i-1}^{E}(t-\delta-1-4 \times d)\right) \\ \\ &\approx & {\sum}_{d=0}^{T}\left( \nabla\vec{\dot \theta}_{i}^{E}(t-\delta-4 \times d)\right) \end{array} $$

(46)

where T = (t − δ)/4.