## 1 Introduction  
Manipulating a virtual object is a fundamental and essential interaction task for a user experiencing augmented reality (AR). To achieve this, selecting, moving, rotating, and changing the size of virtual objects are basic tasks [1]. At least, the selection, movement, and rotation of an (virtual) object, which are used to manipulate a real object in everyday life, should be possible in an AR environment. Currently, owing to the development of computer hardware, several commercial head-mounted displays (HMDs) [2, 3, 4, 5] are available, which a user can use to experience AR in a mobile environment. Providing room modeling using the simultaneous localization and mapping (SLAM) algorithm, some HMDs [3, 4] allow a user to interact with virtual contents [6]. Especially, a virtual interior task [7, 8], which adds virtual furniture in a real room, requires an interaction method for repeatedly and conveniently manipulating the remote virtual object in the AR environment. In such a scenario, there is a limitation in manipulating a virtual object with an existing interface in a mobile AR environment. The touch interaction achieved using the touchpad attached to an HMD [2] provides the user with familiar touch interaction and is advantageous as the user does not need to carry the input device. However, since most of the HMD touchpads are attached to the side of the HMD, the difference between the touch input space and manipulating space makes it difficult and inconvenient for a user to manipulate a virtual object [9]. In contrast, through the hand interaction achieved with a built-in camera in an HMD, the virtual object can be manipulated naturally while the user manipulates a real object. However, because most cameras are attached in front of the HMD, the user must raise his/her hands to interact with a virtual object. When the user repeatedly conducts the task of manipulating the virtual object, this interaction increases the fatigue of the user's arm. The interaction performed using a wand or a controller for manipulating a virtual object [5] has a wider recognition range than the hand interaction and various interactions can be performed with the different interfaces built into the wand. However, this is inconvenient to be used in a mobile environment because the user has to carry additional input devices such as wands and controllers [10]. According to a study on the manipulation of virtual objects [11], manipulating a virtual object with full degree of freedom (DOF) separation, rather than 6 DOF, allows for a more precise control of a virtual object. In addition, using multiple input modalities produces less fatigue than that produced using a single input modality [12; 13]. This paper proposes two interaction methods for conveniently manipulating a virtual object by combining touch interaction and head movements on a mobile HMD for experiencing AR as shown in Figure 1. As the proposed interaction methods consist of two input modalities (touch interaction and head movements), instead of a single input modality, they cause less fatigue when performing a task that iteratively manipulates a virtual object. Moreover, the proposed interaction methods can conveniently manipulate a virtual object because the proposed interaction methods consist of touch interaction, which is widely used in a smartphone or laptop in daily life, and thus, is familiar to a user, and the head movement, which is a frequently used interaction when looking around in everyday life. In this study, we use AnywhereTouch [14] to recognize touch interaction. AnywhereTouch is a method for recognizing the touch interaction with a nail-mounted inertial measurement unit sensor in a mobile environment regardless of the material or slope of the plane [14]. Since AnywhereTouch can recognize the touch interaction regardless of the slope of the plane, it can solve the problems when manipulating a virtual object with a touchpad attached to the HMD.