# American Institute of Mathematical Sciences

September  2019, 9(3): 257-267. doi: 10.3934/naco.2019017

## Bearing rigidity and formation stabilization for multiple rigid bodies in $SE(3)$

 1 Faculty of Science and Engineering, University of Groningen, 9747 Groningen, The Netherlands 2 Department of Control Science and Engineering, Harbin Institute of Technology, 150001, China

* Corresponding author: L. M. Chen

Received  April 2018 Revised  December 2018 Published  May 2019

Fund Project: The first author is supported by China Scholarship Council.

In this work, we first distinguish different notions related to bearing rigidity in graph theory and then further investigate the formation stabilization problem for multiple rigid bodies. Different from many previous works on formation control using bearing rigidity, we do not require the use of a shared global coordinate system, which is enabled by extending bearing rigidity theory to multi-agent frameworks embedded in the three dimensional $special \; Euclidean \; group$ $SE(3)$ and expressing the needed bearing information in each agent's local coordinate system. Here, each agent is modeled by a rigid body with 3 DOFs in translation and 3 DOFs in rotation. One key step in our approach is to define the bearing rigidity matrix in $SE(3)$ and construct the necessary and sufficient conditions for infinitesimal bearing rigidity. In the end, a gradient-based bearing formation control algorithm is proposed to stabilize formations of multiple rigid bodies in $SE(3)$.

Citation: Liangming Chen, Ming Cao, Chuanjiang Li. Bearing rigidity and formation stabilization for multiple rigid bodies in $SE(3)$. Numerical Algebra, Control and Optimization, 2019, 9 (3) : 257-267. doi: 10.3934/naco.2019017
##### References:
 [1] B. D. Anderson, C. Yu and J. M. Hendrickx, Rigid graph control architectures for autonomous formations, IEEE Control Systems Magazine, 28 (2008), 48-63.  doi: 10.1109/MCS.2008.929280. [2] L. Asimow and B. Roth, The rigidity of graphs, Transactions of the American Mathematical Society, 245 (1978), 279-289.  doi: 10.2307/1998867. [3] L. Asimow and B. Roth, The rigidity of graphs, Ⅱ, Journal of Mathematical Analysis and Applications, 68 (1979), 171-190.  doi: 10.1016/0022-247X(79)90108-2. [4] G. Bao and S. Suresh, Cell and molecular mechanics of biological materials, Nature Materials, 2 (2003), 715-725. [5] M. Basiri, A. N. Bishop and P. Jensfelt, Distributed control of triangular formations with angle-only constraints, Systems & Control Letters, 59 (2010), 147-154.  doi: 10.1016/j.sysconle.2009.12.010. [6] A. N. Bishop and M. Basiri, Bearing-only triangular formation control on the plane and the sphere, in 2010 18th IEEE Mediterranean Conference on Control & Automation, 2010 [7] A. N. Bishop, M. Deghat, B. Anderson and Y. Hong, Distributed formation control with relaxed motion requirements, International Journal of Robust and Nonlinear Control, 25 (2015), 3210-3230.  doi: 10.1002/rnc.3250. [8] J. Brewer, Kronecker products and matrix calculus in system theory, IEEE Transactions on Circuits and Systems, 25 (1978), 772-781.  doi: 10.1109/TCS.1978.1084534. [9] R. Connelly, Generic global rigidity, Discrete and Computational Geometry, 33 (2005), 549-563.  doi: 10.1007/s00454-004-1124-4. [10] T. Eren, Using angle of arrival (bearing) information for localization in robot networks, Turkish Journal of Electrical Engineering & Computer Sciences, 15 (2007), 169-186. [11] T. Eren, Formation shape control based on bearing rigidity, International Journal of Control, 85 (2012), 1361-1379.  doi: 10.1080/00207179.2012.685183. [12] T. Eren, W. Whiteley and P. N. Belhumeur, Using angle of arrival (bearing) information in network localization, in 2006 45th IEEE Conference on Decision and Control, (2006), 4676-4681. [13] T. Eren, W. Whiteley, A. S. Morse, P. N. Belhumeur and B. D. Anderson, Sensor and network topologies of formations with direction, bearing, and angle information between agents, in 2003 42nd IEEE Conference on Decision and Control, (2003), 3064-3069. [14] H. Hemmati, Deep Space Optical Communications, John Wiley & Sons, 2006. [15] B. Hendrickson, Conditions for unique graph realizations, SIAM Journal on Computing, 21 (1992), 65-84.  doi: 10.1137/0221008. [16] L. Henneberg, Die Graphische Statik der Starren Systeme, BG Teubner, 1911. [17] G. Laman, On graphs and rigidity of plane skeletal structures, Journal of Engineering Mathematics, 4 (1970), 331-340.  doi: 10.1007/BF01534980. [18] G. Michieletto, A. Cenedese and A. Franchi, Bearing rigidity theory in SE (3), in 2016 55th IEEE Conference on Decision and Control, (2016), 5950-5955. [19] B. Roth, Rigid and flexible frameworks, The American Mathematical Monthly, 88 (1981), 6-21.  doi: 10.2307/2320705. [20] E. Schrijver and J. Van Dijk, Disturbance observers for rigid mechanical systems: Equivalence, stability, and design, Journal of Dynamic Systems, Measurement, and Control, 124 (2002), 539-548. [21] T.-S. Tay and W. Whiteley, Generating isostatic frameworks, Structural Topology, 11 (1985), 21-69. [22] R. Tron, J. Thomas, G. Loianno, K. Daniilidis and V. Kumar, A distributed optimization framework for localization and formation control: Applications to vision-based measurements, IEEE Control Systems Magazine, 36 (2016), 22-44.  doi: 10.1109/MCS.2016.2558401. [23] D. Zelazo, A. Franchi and P. R. Giordano, Rigidity theory in SE (2) for unscaled relative position estimation using only bearing measurements, in 2014 13nd European Control Conference, (2014), 2703-2708. [24] D. Zelazo, P. R. Giordano and A. Franchi, Bearing-only formation control using an SE (2) rigidity theory, in 2015 54th IEEE Conference on Decision and Control, (2015), 6121-6126. [25] S. Zhao and D. Zelazo, Translational and scaling formation maneuver control via a bearing-based approach, IEEE Transactions on Control of Network Systems, 4 (2017), 429-438.  doi: 10.1109/TCNS.2015.2507547. [26] S. Zhao and D. Zelazo, Bearing rigidity and almost global bearing-only formation stabilization, IEEE Transactions on Automatic Control, 61 (2016), 1255-1268.  doi: 10.1109/TAC.2015.2459191.

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##### References:
 [1] B. D. Anderson, C. Yu and J. M. Hendrickx, Rigid graph control architectures for autonomous formations, IEEE Control Systems Magazine, 28 (2008), 48-63.  doi: 10.1109/MCS.2008.929280. [2] L. Asimow and B. Roth, The rigidity of graphs, Transactions of the American Mathematical Society, 245 (1978), 279-289.  doi: 10.2307/1998867. [3] L. Asimow and B. Roth, The rigidity of graphs, Ⅱ, Journal of Mathematical Analysis and Applications, 68 (1979), 171-190.  doi: 10.1016/0022-247X(79)90108-2. [4] G. Bao and S. Suresh, Cell and molecular mechanics of biological materials, Nature Materials, 2 (2003), 715-725. [5] M. Basiri, A. N. Bishop and P. Jensfelt, Distributed control of triangular formations with angle-only constraints, Systems & Control Letters, 59 (2010), 147-154.  doi: 10.1016/j.sysconle.2009.12.010. [6] A. N. Bishop and M. Basiri, Bearing-only triangular formation control on the plane and the sphere, in 2010 18th IEEE Mediterranean Conference on Control & Automation, 2010 [7] A. N. Bishop, M. Deghat, B. Anderson and Y. Hong, Distributed formation control with relaxed motion requirements, International Journal of Robust and Nonlinear Control, 25 (2015), 3210-3230.  doi: 10.1002/rnc.3250. [8] J. Brewer, Kronecker products and matrix calculus in system theory, IEEE Transactions on Circuits and Systems, 25 (1978), 772-781.  doi: 10.1109/TCS.1978.1084534. [9] R. Connelly, Generic global rigidity, Discrete and Computational Geometry, 33 (2005), 549-563.  doi: 10.1007/s00454-004-1124-4. [10] T. Eren, Using angle of arrival (bearing) information for localization in robot networks, Turkish Journal of Electrical Engineering & Computer Sciences, 15 (2007), 169-186. [11] T. Eren, Formation shape control based on bearing rigidity, International Journal of Control, 85 (2012), 1361-1379.  doi: 10.1080/00207179.2012.685183. [12] T. Eren, W. Whiteley and P. N. Belhumeur, Using angle of arrival (bearing) information in network localization, in 2006 45th IEEE Conference on Decision and Control, (2006), 4676-4681. [13] T. Eren, W. Whiteley, A. S. Morse, P. N. Belhumeur and B. D. Anderson, Sensor and network topologies of formations with direction, bearing, and angle information between agents, in 2003 42nd IEEE Conference on Decision and Control, (2003), 3064-3069. [14] H. Hemmati, Deep Space Optical Communications, John Wiley & Sons, 2006. [15] B. Hendrickson, Conditions for unique graph realizations, SIAM Journal on Computing, 21 (1992), 65-84.  doi: 10.1137/0221008. [16] L. Henneberg, Die Graphische Statik der Starren Systeme, BG Teubner, 1911. [17] G. Laman, On graphs and rigidity of plane skeletal structures, Journal of Engineering Mathematics, 4 (1970), 331-340.  doi: 10.1007/BF01534980. [18] G. Michieletto, A. Cenedese and A. Franchi, Bearing rigidity theory in SE (3), in 2016 55th IEEE Conference on Decision and Control, (2016), 5950-5955. [19] B. Roth, Rigid and flexible frameworks, The American Mathematical Monthly, 88 (1981), 6-21.  doi: 10.2307/2320705. [20] E. Schrijver and J. Van Dijk, Disturbance observers for rigid mechanical systems: Equivalence, stability, and design, Journal of Dynamic Systems, Measurement, and Control, 124 (2002), 539-548. [21] T.-S. Tay and W. Whiteley, Generating isostatic frameworks, Structural Topology, 11 (1985), 21-69. [22] R. Tron, J. Thomas, G. Loianno, K. Daniilidis and V. Kumar, A distributed optimization framework for localization and formation control: Applications to vision-based measurements, IEEE Control Systems Magazine, 36 (2016), 22-44.  doi: 10.1109/MCS.2016.2558401. [23] D. Zelazo, A. Franchi and P. R. Giordano, Rigidity theory in SE (2) for unscaled relative position estimation using only bearing measurements, in 2014 13nd European Control Conference, (2014), 2703-2708. [24] D. Zelazo, P. R. Giordano and A. Franchi, Bearing-only formation control using an SE (2) rigidity theory, in 2015 54th IEEE Conference on Decision and Control, (2015), 6121-6126. [25] S. Zhao and D. Zelazo, Translational and scaling formation maneuver control via a bearing-based approach, IEEE Transactions on Control of Network Systems, 4 (2017), 429-438.  doi: 10.1109/TCNS.2015.2507547. [26] S. Zhao and D. Zelazo, Bearing rigidity and almost global bearing-only formation stabilization, IEEE Transactions on Automatic Control, 61 (2016), 1255-1268.  doi: 10.1109/TAC.2015.2459191.
Comparison of different definitions for bearing and bearing rigidity
 Definitions for bearing Measurement variable Rigidity Angle in 2D space $\theta_{ij}$ Parallel bearing rigidity Unit vector in a global frame $\frac{p_j-p_i}{||p_j-p_i||}$ Bearing rigidity in $\mathbb{R}^d$ Unit vector in $SE(2)$ $T(\theta_i)\frac{p_j-p_i}{||p_j-p_i||}$ Bearing rigidity in $SE(2)$
 Definitions for bearing Measurement variable Rigidity Angle in 2D space $\theta_{ij}$ Parallel bearing rigidity Unit vector in a global frame $\frac{p_j-p_i}{||p_j-p_i||}$ Bearing rigidity in $\mathbb{R}^d$ Unit vector in $SE(2)$ $T(\theta_i)\frac{p_j-p_i}{||p_j-p_i||}$ Bearing rigidity in $SE(2)$
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