I am currently designing a symmetric parallel mechanism that can be modeled as follows:

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Here, $\tau^j$ denotes actuator moments and $k_i$ are torsion springs constants. For a specific point in the workspace of the system, the actuator moments as well as the torsion angles $\varphi_i^j$ are known. It is of interest to determine possible sets of torsion spring constants $\mathbf{k}$ that would result in the torsion angles $\boldsymbol{\varphi}$, given the actuator moments $\boldsymbol{\tau}$. The links can be assumed rigid.

In other words, I am looking for the relationship between the actuator moments, torsion spring constants and torsion angles when the mechanism is in static equilibrium.

My current approach is to look at the total energy of the system. Since the potential energy of a torsion spring is $$ U = \frac{1}{2}k\varphi^2 = \frac{1}{2}\tau\varphi, $$ the total energy of the system in static equilibrium can be written as $$\frac{1}{2}(\tau^1\varphi_1^1 + \tau^2\varphi_1^2 + \tau^3\varphi_1^3) = \frac{1}{2}(k_1(\varphi_1^1)^2 + k_1(\varphi_1^2)^2+k_1(\varphi_1^3)^2+k_2(\varphi_2^1)^2 + k_2(\varphi_2^2)^2+k_2(\varphi_2^3)^2+k_3(\varphi_3^1)^2 + k_3(\varphi_3^2)^2+k_3(\varphi_3^3)^2)$$ or $$ \sum_{j=1}^3\tau^j\varphi_1^j = \sum_{i=1}^3\sum_{j=1}^3k_i(\varphi_i^j)^2. $$

Is this the right way to model this system, or is my approach omitting some important observation?


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