## 1 Introduction
The pneumatic muscle (PM) [16; 17] was developed in the 1950s and is suited to devices that are worn. The retraction strength of a PM depends on the strength of the individual fibers in the woven shell. A PM has several advantages in that it is highly compliant, low-cost, lightweight, and safe. It is used in rehabilitation engineering, which requires great compliance and a high level of safety for patients [18]. A study by [PERSON] et al. [19] demonstrates a pneumatic-muscle ASS that uses one FESTO artificial muscle (MAS-20-N400) parallel with two springs to provide a vertical force for vehicles. The study shows that the PM-driven ASS can vibrate at a 40 mm-amplitude at the frequency of 1.5 Hz and vibrates at 10-25 mm-amplitude within the frequency range of 4-8 Hz. The length of the PM is 400 mm and air flows into the PM from one side only. The report demonstrates that the main challenge in designing a PM-driven ASS is the inherent non-linear deformation characteristics. A PM-ASS has never used in commercial vehicles; thus, this study develops a test-rig for a PM-driven ASS and determines its ability to reduce vibration. The test-rig is designed as a quarter-car and comprises a FESTO PM, which is installed between the sprung mass and the wheel and two MacPherson struts provide additional vertical force. The test-rig simulates different driving conditions in terms of the speed of rotation of the wheel and road terrain. For ASS control, an adaptive Fourier-series-based adaptive sliding-mode controller (FSB-ASMC) with \(H_{\infty}\) tracking performance is used to compensate for all possible nonlinearities, uncertainties, and external vibration. A Fourier neural network (Fourier-NN) [20; 21] combines Fourier analysis and neural network theory. A Fourier-NN uses a family of complex Fourier functions as an activation function \(\sigma_{i}(\cdot)\), which decomposes a signal into its component frequencies with different amplitudes and phases. The FSB-ASMC also uses \(H_{\infty}\) tracking for the adaptive sliding-mode control scheme, which protects the systems against approximation errors, disturbances and unmodeled dynamics and guarantees the desired \(H_{\infty}\) tracking performance for the overall system. It also significantly reduces the control chatter that is inherent in a conventional sliding mode control system. The main contributions of this paper are as follows:  
1. This paper uses a pneumatic muscle actuator with two MacPherson struts to increase the isolation of vibration and show its feasibility on a quarter-car test rig. 2. A fully-functional test-rig for a quarter car is developed for examining the PM-ASS which has a road profile generator to provide real road conditions. 3. Using the orthogonal activation basis functions of Fourier series, Fourier-NN has a clear physical meaning and readily determined structure, which gives it an advantage over conventional NNs. Because of the orthogonality of the basis functions, the Fourier-NN converges very fast, which means it is suited to real-time implementation. 4. A state-predictor \(\mathit{GM}(1,1)\) is used to predict the output error and to provide the predictions for the controller. 5. Four different control strategies are conducted to analyze the stability and accuracy in terms of reducing vibration, which are MacPherson Strut, FSASMC, FSB-ASMC+ \(H_{\infty}\), and GP-FSB-ASMC+ \(H_{\infty}\).