## 1 Introduction  
Buttons serve as a critical medium for user-device interaction, primarily facilitating control and input functions. They find widespread application in our daily lives, allowing us to remotely control various devices, such as adjusting television volume or setting refrigerator temperatures through electronic screens. Both virtual and physical buttons, as integral elements of a product, require careful consideration of their spatial placement and interrelationships within the product to meet user operational needs and enhance efficiency. The act of pressing buttons represents the direct form of user-product interactions. To enhance comfort and efficiency, button layout design should incorporate human hand size and movement patterns. This approach ensures that the buttons are easily accessible and user-friendly, optimizing the overall user experience with the product [1]. For virtual buttons, size and position significantly influence user preference [2]. Button layouts can be categorized as two-dimensional (2D) or three-dimensional (3D). Two-dimensional layouts are commonly found in touchscreens and control panels, and consideration of thumb reach on touchscreens can enhance user experience [3]. Control panel operation modes are determined by the product's functionality and form, with extensive research in human-machine analysis based on static human torso sizes and dynamic joint movements to derive design references for specific applications [4]. For 3D layouts, which are in prolonged contact with the human body, ergonomic dimensions and aesthetics must be thoroughly considered. Due to their varied shapes and operational modes, creating a unified design principle for button layouts is challenging. Ergonomic methods are the primary basis for design, utilizing standard human body measurementsto align with physiological characteristics. Research has identified optimal layouts for automotive electronic gear shift buttons among 12 options through ergonomic subjective evaluations [5]. In the case of split keyboards, adjustments like keyboard tilt angle and wrist rest height enhance overall button layout operability, improving user experience [6]. The joystick button layout is a typical example of the 3D button layout design. Joysticks convey user commands and are presented in standalone or fixed forms on large equipment as input devices. They are usually equipped with one or more buttons to transmit signals and also respond to joystick angles or directions. Joysticks are used extensively in engineering machinery, transportation, aerospace, medicine, and entertainment. Their operation mode remains stable and uniform, necessitating users to maintain a full-contact grip. Therefore, designing the joystick's form and button configuration must comprehensively consider ergonomic factors. The joystick's overall size [7], shape [8], and placement [9] are critical factors affecting operational performance, user physical load, and psychological assessment. Early studies employed reverse modeling of human grip to create joysticks that fit the gripping action [10; 11]. Handheld devices and handles share similar hand operation requirements with joysticks, and their button layouts significantly affect user experience. Research on the Switch game controller found that smaller button distances can reduce comfort and gaming experience [12]. Another study [13] developed an ergonomic handle for medical surgery scenarios based on tool gripping posture, designing two operational buttons according to finger distribution and dynamic actions during gripping. While extensive research has applied ergonomic methods to design and optimize button layouts for complex 3D models, joystick button layout design still faces limitations. On the one hand, many button layouts are still based on confirmed shapes. Although the overall shape and size of joysticks have been scientifically validated, button placement often relies on subjective designer judgments, resembling arbitrary placement rather than quantitative assessment.