The inverted Pendulum: A fundamental Benchmark in Control Theory and Robotics Olfa Boubaker National Institute of Applied Sciences and Technology INSAT, Centre Urbain Nord BP. 676 – 1080 Tunis Cedex, Tunisia  olfa.boubaker@insat.rnu.tn  Abstract —   For at least fifty years, the inverted pendulum has been the most popular benchmark, among others, for teaching and   researches   in   control   theory   and   robotics.   This   paper presents the key motivations for the use of that system and explains, in details, the main reflections on how the inverted pendulum benchmark gives an effective and efficient application. Several real experiences, virtual models and web-based remote control   laboratories will   be   presented   with   emphasis   on   the practical design implementation of this system. A bibliographical survey of different design control approaches and trendy robotic problems will be presented through applications to the inverted pendulum system. In total, 150 references in the open literature, dating back to 1960, are compiled to provide an overall picture of historical, current and challenging developments based on the stabilization principle of the inverted pendulum.  Keywords-   Inverted pendulum, Control theory, Robotics.  I.   I   NTRODUCTION Control theory is a field rich in opportunities and new directions [1] dealing with disciplines and methods that leads to an   automatic   decision   process   in   order   to   improve   the performance of a control system. The evolution of control theory   is   related   to   research   advances   on   technology, theoretical   controller   design   methods   and   their   real-time implementation [2, 3]. It is important to note that evolution of control theory is also closely related to education [4, 5]. The next generations of control students must receive the scientific and   pedagogical   supports   required   to   verify   conventional techniques, develop new tools and techniques and verify their realization [6]. If the student has no opportunity to bring into the   light   with   realistic   control   problems   subjected   to   the difficulties, training has failed [7].   The main recommendation is   then   to   integrate   into   the   research   laboratories   and curriculum education experimental projects to bridge the gap between   the   acquirement   of   scientific   knowledge   and   the solution of practical problems. In recent Years,   projects on the themes of robotics [8, 9] and mechatronics [8, 10, 11, 12] are the most attractive for students.   In   this   framework,   many   interesting   robotic benchmark   systems   exist   in   the   literature.   The   inverted pendulum system was always considered, among others, the most fundamental benchmark. Different versions of this system exist offering a variety of interesting control challenges. This   paper,   proposes   to   enhance   the   wealth   of   this benchmark   and   attempt   to   provide   an   overall   picture   of historical, current and trend developments in control theory and robotics based on its simple structure. The paper is organized as follows: The most common robotic benchmarks and the different versions of the inverted pendulum will be exposed in the next section. The wealth of the inverted pendulum model in education will be pointed in section 3.   In section 4, few real and   virtual experiences   will   be exposed.   Different control design techniques will be surveyed through application to this benchmark in section 5. Trendy robotic problems based in the inverted pendulum stabilization principle will be finally raised in section 6. II.   R OBOTIC   B   ENCHMARKS :   A N   O   VERVIEW Many robotic benchmark systems of high interest exist in the literature and frequently used for teaching and research in control theory. They are typically used to realize experimental models, validate the efficiency of emerging control techniques and verify their implementation.   The most common robotic benchmarks are the Acrobot [13], the Pendubot [14], the Furuta Pendulum (see Fig.1) [15], the inverted pendulum [16], the Reaction Wheel Pendulum (see Fig.2) [17], the bicycle [18], the VTOL aircraft [19], the Beam-and-Ball system [20] and the TORA [21].  Figure 1. Furuta Pendulum In spite of its simple structure, the inverted pendulum is considered, among the last examples, the most fundamental benchmark.   Different   versions   of   the   inverted   pendulum benchmark   exist   offering   a   variety   of   interesting   control challenges.
Figure 2.Inertia-Wheel Pendulum The   most   familiar   types   are   the   rotational   single-arm pendulum (see Fig.3) [22], the cart inverted pendulum (see Fig.4) [23], and the double inverted Pendulum [24].  Fig.3. Rotational single-arm pendulum The   less   common   versions   are   the   rotational   two-link pendulum [25], the parallel type dual inverted pendulum [26], the   triple   inverted   pendulum   [27],   the   quadruple   inverted pendulum [28] and the 3D or spherical pendulum [29].  Figure 4. Cart inverted pendulum III.   T   HE INVERTED   P   ENDULUM   B   ENCHMARK IN   C ONTROL T   HEORY   E   DUCATION Since   the   1950s,   the   inverted   pendulum   benchmark, especially   the   cart   version,   was   used   for   teaching   linear feedback control theory [30] to stabilize open-loop unstable systems. The first solution to this problem was described by Roberge in 1960 [31] and then by Schaefer, and Cannon in 1966 [32]. This benchmark was considered in many references as a typical root-locus analysis example [33]. Subsequently, it has been also used in many books to solve the linear optimal control   problem   [34]   and   the   complex   nonlinear   control problem [35] for unstable systems. In   spite   of the   simplicity of   its   structure,   an inverted pendulum   system   is   a   typical   nonlinear   dynamic   system including a stable equilibrium point when the pendulum is at pending position and an unstable equilibrium point when the pendulum is at upright position. When the system is moved up from the pending position to the upright position, the model is strongly nonlinear with the pendulum angle. The principal control task considered for the cart pendulum version   is   to   swing   up   the   pendulum   from   the   stable equilibrium point to the unstable equilibrium point, and then balance the pendulum at the upright position, and further move the cart to a specified position by driving it right and left. The more   general problem is guiding the pendulum from any arbitrary   initial   condition   to   the   upright   equilibrium   and stabilizes the cart in a desired position. IV.   R EAL DEVICES ,   VIRTUAL MODELS AND WEB   - BASED LABORATORIES The simple structure of the inverted pendulum model allowing carrying   out   experimental   validations   is   one   of   the   key motivations   of   using   this   benchmark   in   education   and   in research. Several real devices, virtual models or web based remote control laboratories are then developed.  •   Real control laboratories  Many real experimental models of the inverted pendulum are performed allowing to students and researches to validate the efficiency of several emerging control techniques and verify their implementation [36, 37, 38]. In many cases, common integrated software’s like Matlab or LabVIEW are used to support the design of controllers, the analysis of control system and also the real-time implementation of controllers [39, 40, 41].  •   Virtual reality models  Today, physical models might be considered old fashioned, unreliable and expensive. There are many reasons to replace them by virtual reality enabled by the computing power of computers and software tools. One may find good examples of virtual control Laboratories of pendulum-car benchmark in [42, 43, 44].
•   Web-based laboratory for remote control  Distance education as well as remote experimentation, is a modern field that aims to deliver education to students who are not   physically   present   on   their   tested   instrumentation. Nowadays, many students communicate with the educational material through Internet-based technologies. Time savings, sharing   resources   of   expensive   equipment   and   individual access to experimentation are the main factors that motivate remote laboratories over the world. Remote pioneered projects based on the inverted pendulum can be found in [45, 46, 47, 48]. V.   T   HE INVERTED   P   ENDULUM   B   ENCHMARK IN   C ONTROL T   HEORY   R ESEARCH In control theory, a rich collection of powerful and successful synthesis   methods   are   available   today   for   which   many academic   books   and   survey   papers   are   devoted.   In   this framework, the inverted pendulum has maintained for at least fifty years its usefulness to illustrate almost all emerging ideas in this field (see Table 1). TABLE I.   S URVEY OF   C   ONTROL   D ESIGN   T ECHNIQUES   I LLUSTRATED BY THE   I NVERTED   P ENDULUM Control design   Academic Books   Survey papers   Research papers with illustration Bang-Bang control   -   [49]   [50, 51] Fuzzy logic control   [52]   [53, 54]   [55,56] Neural   Network control [57, 58]   [59]   [60] PID Adaptive control   [61]   [62, 63, 64, 65]   [66] Robust control   [67, 68, 69]   [70,71]   [72] Energy based control   [35,73,74,75]   [76, 77]   [22,78,79,80,81,82, 83, 84,85] Hybrid control   [86, 87, 88]   [89,90,91,92, 93]   [38,94, 95,96,97] Sliding mode control   [98,99,100]   [101,102]   [103,104,105] Time optimal control   106, 107, 108]   [109]   [110,111,112,113] Predictive control   [114,115,116,117,118]   [118,119,120,121]   [122,123,124,125] Singular perturbation   [126]   [127, 128]   [129] Feedforward control   -   [130]   [131, 132] VI.   T   RENDS IN   R OBOTICS BASED ON THE   I   NVERTED P   ENDULUM   S TABILIZATION   P   RINCIPLE Based on the inverted pendulum stabilization principle, many trendy technologies in robotics will fertilize quite new control applications. The recent major accomplishments are:  •   Control of under-actuated robotic systems  Underactuated   robotic   systems   are   systems   with   fewer independent control actuators than degrees of freedom to be controlled. In recent years, the need for analysis and control of underactuated   robotic   systems   arises   in   many   practical applications [79, 133, 134]. The cart inverted pendulum is a typical system of under-actuated robotic systems [135].  •   Design of mobile inverted pendulums  Design and implementation of mobile wheeled inverted pendulum systems have induced a lot of attention recently [136,137,138]   and   at   least   one   commercial   product,   the Segway, see Fig.5, is offered [139]. Such vehicles are of interest because they have a small trail.  Fig.5. The Segway: A mobile wheeled inverted pendulum [139]  •   Gait pattern generation for humanoid robots  The control of humanoid robots [140] is a challenging task in recent years due to the hard-to-stabilize dynamics. Gait pattern generation is a key problem [141, 142]. In order to simplify the trajectory generation, many studies make use of the analogy between   bipedal gait   and the inverted pendulum  motion [143].   The Linear Inverted Pendulum Model (LIPM) [144, 145, 146, 147, 148, 149] is a trendy research topic in this framework. LIPM algorithms are generally joined to Zero Moment Point concept [150] to ensure stable gait (see Fig.6).  Fig.6. HRP-4C Humanoid: stable walking under linear inverted pendulum model [150] VII. C ONCLUSION In this paper, it has been shown that the inverted pendulum system is a fundamental benchmark in education and research in control theory. 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